### Pointless Gigantic List of Numbers - Part 3 (10^10^1,000,000 ~ order type w)

 (back to part 2) PART 3: THE FRIENDLY PRELIMINARIES 10^10^1,000,000 ~ fw(n) numbers This part starts with a millionduplex, ascends past a googolduplex and other tetrational numbers, and quickly reaches bigger unfamiliar numbers. As big as they are, all these numbers are small enough to represent or approximate with most of popular large number notations: exponents, tetration, the Ackermann function, up-arrow notation, and the beginning of Conway chain arrows. The Fourth-Class Range 10^10^1,000,000 ~ 10^10^10^1,000,000 Entries: 35   Millionduplex 10^10^1,000,000 The boundary between class 3 and 4 numbers. Class 4 numbers range from 10^10^1,000,000 (E6#3) to 10^10^10^1,000,000 (E6#4) - they can't be approximated accurately, but their number of digits can be. They are numbers when 2*n can no longer really be distinguished from n, and begin a stage where the numbers begin to mess with your mind (see Skewes' Approxima for instance). Both of Skewes' numbers and a googolduplex are examples of class 4 numbers. In my opinion, class 4 numbers are a major turning point in large numbers: they transcend approximations that can be considered accurate in ordinary terms, and also are a good near-transcension of the real world. After all, even the small class 4 numbers transcend a promaxima, which can be argued to be the largest meaningful number probability-wise. Fzmilliplexion (10^1,000,000)^(10^1,000,000) = 10^10^1,000,006 This is a number Sbiis Saibian's brother came up with in an attempt to show that naming large numbers is always easy. Sbiis Saibian himself gave this number a name, fzmilliplexion, from the fz- prefix which takes a number to the power of itself and milliplexion, a name he gives for ten to the millionth power. This number is equal to the millionth power of a millionduplex, but it turns out that at this scale that isn't much of an improvement. Hold on now, how can that be right?! Just observe: (10^1,000,000)^(10^1,000,000) = 10^(1,000,000*10^1,000,000) - we're just using the exponential laws taught in algebra here! = 10^(10^6*10^1,000,000) = 10^(10^(1,000,000+6)) = 10^10^1,000,006 It's all about the counterintuitivity of hyper-exponential functions - you need to throw away your usual number sense because it's really misleading with numbers this big, and bring out a googology sense. However this is a very tame example compared to the even bigger giants! Number of ways to arrange the books in the Library of Babel ~ 10^10^1,834,103 This is an example of a number "way out there" that we can generate using the realm of probability, but is debatable whether it's "phyiscally meaningful". It's the number of ways to arrange the books in the Library of Babel, a fictional library consisting of every possible book with 410 pages, 3200 characters per page, and 25 different possible characters (the number of books in it is known as Borges' number, see that entry for more on the Library of Babel). By taking the factorial of Borges' number, we get the number of ways to arrange the books in that library. There are 1.83 million digits in the number of digits in this number. 8^8^8^8 8^^4, ~10^10^15,151,335 This is 8 tetrated to the fourth, a number with a 15-million-digit number of digits. 10^10^33,014,740 A commonly cited erroneous value of the number of books to arrange the Library of Babel, from the book "The Unimaginable Mathematics of the Library of Babel". Googolplexibong 10^10^100,000,000 Can be written as E8#3 in Hyper-E notation. The value is much larger than 8^^4 but much less than 9^^4. 9^9^9^9 9^^4, ~10^10^369,693,099 This is 9 tetrated to the fourth or 9^9^9^9. Like its cousin 9^9^9, this number has shown up a lot in large number discussions, being the largest number you can name with 4 digits and standard mathematical operators. Scott Aaronson mentioned this number several times in his essay "Who Can Name the Bigger Number" as an example of how easy it is to name very large numbers. Tetralogue 10^^4 = 10^10^10^10 This number is equal to one followed by a trialogue zeros, a.k.a. one followed by "one followed by ten billion zeros" zeros. These numbers are getting close to numbers that can’t represent anything in the real world. But this is still less than numbers like Poincare recurrence times. This is the fourth member of the -logue series, one followed by "one followed by ten billion zeros" zeros, an unfathomable sized number. Googolplexithrong 10^10^10^11 Can be written E11#3 in Hyper-E. The number is between a tetralogue and Skewes' Number. 3^3^3^3^3 ~ 10^10^(3.638*10^12) This is three tetrated to the fifth. It's notable for being the smallest power tower of threes bigger than a googolplex, and it's commonly compared to the much larger tritri, equal to 3 pentated to 3, a.k.a. a power tower of 7.6 trillion threes. There are 3.6 trillion digits in the number of digits in this number. Hollom's odds that all parallel universes are the same ~10^10^10^16 In the second part of Lawrence Hollom's website, he discusses large numbers that occur in probability such as the monkey-typewriter number and other similar bigger numbers. He finishes off with what he claims is the largest number that can represent anything in the real world: Hollom says that with a promaxima, the number of possible different parallel universes assuming sub-Planck units aren't meaningful, and the number of parallel universes out there, we can calculate the odds that all those universes are the same. He gives 10^10^16 as an estimate of the number of parallel universes in the multiverse (although that is actually an estimate of the number of distinguishable parallel universes), and then takes the old value of the promaxima (10^10^245) to the power of 10^10^16, and calculates that to be approximately 10^10^10^17, even though it's really about 10^10^10^16. Though this number may be the largest number that can represent real-world things probability-wise, Hollom clearly hadn't heard of Poincare recurrence times. f3(4) ~ 10^10^10^20.5506 This is a HUGE number equal to f3(4) in thee fast-growing hierarchy. This monstrous number falls between a googolplexithrong and Skewes' number, making it a good tetrational number. However, it's a very tame example of what the fast-growing hierarchy has to offer! f3(x) achieves tetrational growth, but f4(x) crushes that with pentational growth! Skewes’ Number e^e^e^79 ~ 10^10^(8.85214*10^33) Main article: Skewes' numbers Skewes' number is one of the "classic" large numbers, along with the googol and googolplex, Graham's number, the Steinhaus-Moser notation numbers, and some others. It was defined by Stanley Skewes in a 1933 mathematical proof, as an upper-bound for the first point where the prime counting function (noted π(x), the number of prime numbers below x) crosses the logarithmic integral function (noted li(x), defined as the integral:     x     /\     |     |  1/ln(t) dt     |    \/     0 assuming the Riemann hypothesis to be true. The number falls between a tetralogue and a googolduplex, and at one time it was the largest number used in a mathematical proof before the second Skewes' number came along - the larger second Skewes' number was proven as an upper-bound without assuming the Riemann hypothesis about 20 years later. Since Skewes' time the upper-bound for the problem has been improved significantly - the current upper-bound is 1.39*10^316, still a sizable number and reasonably close to a centillion. The current lower bound is 10^14 (100 trillion). But nonetheless, due to its surprising size (see my article on Skewes' numbers for a detailed coverage of how big it is) it remains a famous large number. Skewes' number is probably the fourth most famous googolism, after the googol and googolplex (tied for first) and Graham's number (third). Skewes’ Approxima 10^10^10^34 Sbiis Saibian explained this way better than I could, so I will quote him on his ultimate large number list:           This is an approximation usually given for Skewes' Number, since Skewes' Number is about 10^10^10^33.947. This might seem like a good approximation, but this value is actually A LOT LARGER THAN Skewes' Number. How much larger? You'd have to raise Skewes' Number to the power of about 10^10^33 to get 10^10^10^34!            Here's a way to get an idea of what that means. Imagine that you had a sphere containing roughly a Skewes' Number particles. That sphere would be massive, even assuming the particles were tightly packed. Now imagine that sphere being just one amongst a Skewes' Number of such spheres! Imagine all these spheres are contained in A 2nd order "Skewes' sphere". Now imagine that is only one amongst a Skewes' Number of 2nd order Skewes' spheres all contained in a 3rd order Skewes' sphere!! Now keep scaling up to the 4th order, 5th, 6th, 7th, 8th, 9th, 10th, ... 100th, 1000th, millionth, billionth, trillionth, ... centillionth, ... ... ... ... and keep on going until you reach the 10^10^33 order sphere. That sphere will contain roughly 10^10^10^34 particles! Mind boggling! And this is only the difference between 10^10^10^33.947 and 10^10^10^34, and we're still only talking about moderately sized tetrational numbers!!! In other words, looks are REALLY deceiving with numbers this big ... but you know what they say: it only gets worse! Poincare Recurrence Time (low) 10^10^10^76 There is a theory that states that the universe will return to its previous state after a long but finite time. This is that length of time for a black hole of stellar mass. Poincare recurrence times are the largest numbers to have occurred in any science! Googolyllion 10^(2^10^100+2) ~ 10^10^10^99.4786 This is the -yllionized version of a googol, or the googolth -yllion, since the xth -yllion is equal to 10^(2^n+2) (see 100,000,000). It's an amusing number analogous to the googolillion. It's "slightly" smaller than a googolduplex. Googolduplex 10^10^10^100 A googolduplex is 1 followed by a googolplex zeros. It's also known as a googolplexplex or a googolplexian (or erroneously as a gargoogolplex which is actually the square of a googolplex). However, among the googology community there is general consensus that this number should be called a "googolduplex", mostly because its name is systematic and -duplex is a contraction of -plexplex and literally means "two plex". However, although I highly prefer googolduplex I have also seen supporters of the name "googolplexian" over "googolduplex" in the googology community. A googolduplex was probably originally created as a response to the googolplex, and has been further extended to a googoltriplex, quadriplex, quintiplex, etc, using Latin roots. A googolduplex is at a stage where the numbers are difficult to compare to real world values - that is typical of class 4 numbers (see 10^10^1,000,000). This number can be written E100#3 or E2#4 in Hyper-E notation. It's a lower bound to a googolplex-bang since 10^x is a lower bound for x! when x is at least 25. A googolduplex would take a number of letters on the scale of a googol to write out in terms of any -illions (i.e. it's hopelessly impossible), but here's how it would start and end in terms of Bowers' -illions: ten tretractratriotriacontetriotriahecto............................... ...............................tretrigintitrecentimilli-duotrigintitrecentillion It's pretty hard to try to start representing a googolduplex in terms of Bowers' intricate -illion system, and the number has a very intricate name for one, using hundreds of different roots. In Conway and Guy's system, the name is far more mundane though: ten trillitrestrigintitrecentillitrestrigintitrecentillitrestrigintitrecentilli........................ .....................trestrigintitrecentillitrestrigintitrecentillitrestrigintitrecentilliduotrigintitrecentillion where trestrigintitrecentilli is repeated about a third of a googol times. The name there has exactly eight-googol-and-one letters. (pointless fact: the guy who madeis a fairly vehement googolplexian hater which is pretty cool) Googolplex-bang (10^10^100)! The factorial of a googolplex. This is “slightly” larger than than the googolduplex and “slightly” smaller than taking a googolplex to its own power. Fzgoogolplex (10^10^100)^(10^10^100) This is a googolplex to the power of a googolplex, a classic example of a layman's attempt to name the biggest number they can. Tim Urban mentioned this number on his blog post "From 1,000,000 to Graham's Number", saying that before he learned about Graham's number he thought that numbers like a googolplex to the googolplexth power were the biggest numbers a human could conceive of. As it turns out this number does NOT EVEN COME CLOSE to Graham's number, although Graham's number is itself NOWHERE CLOSE to the largest numbers a human could conceive of. This number evaluates to: (10^10^100)^(10^10^100) = 10^(10^100*10^10^100) = 10^10^(10^100+100). That last form looks only a tad bigger than a googolduplex, but it's actually equal to raising it to the googolth power! This means that if you were to start with a googolduplex, and dwarf it by a factor of itself every Planck time (that means you would dwarf it by a factor of itself 1043 times every second!), it would take an unimaginably long 1.706*1049 years to get a fzgoogolplex!! 10^10^10^101 A lower bound for googolbang-plex, googoldubang, and fugagoogol. Googolbang-plex 10^((10^100)!) Between 10^10^10^101 and 10^10^10^102. It is of note that even though x-bang is more powerful than x-plex, banging the googolplex is less powerful than plexing the googolbang. Googolbangbang / googoldubang ((10^100)!)! A number I made up as the factorial of the googol-bang; on the logarithmic level it’s indistinguishable from the googolbang-plex. Googolbangbang is the obvious name, but googoldubang sounds much nicer, just like googolplexplex vs. googolplexian vs. googolduplex. Fugagoogol Googol vv Googol Each v is a down arrow. Down-arrow notation is like up-arrow notation, but evaluated from left to right instead of right to left; therefore it is less powerful. If you don’t get it here’s a quick definition: x v y = x^y x vv y = ((x v x) v x)....v x)...)) with y x’s x vvv y = ((x vv x) vv x)....vv x)...)) with y x’s etc. This number is about 10^10^10^102, exactly 10^10^(10^102-98). It looks slightly larger than a googolduplex, but the looks are deceiving - the number of digits is actually about a googolplex, raised to the 100th power! (for more on this number look here) 10^10^10^102 An upper-bound on googolbang-plex, googoldubang, and fugagoogol. You can see that there are various functions that can be used to create numbers in the neighborhood of a googolduplex. Poincaré Recurrence Time (mid) 10^10^10^120 This is the value of a Poincaré recurrence time for a black hole with mass of the observable universe. Cosmological inflationary size of the universe 10^10^10^122 If the universe were to expand as much as it did when it was born, it would be this big. That’s over a googolduplex! This number is so large that it makes no difference whether we use Planck lengths or yottameters. Ecetonduplex 10^10^10^303 Continuing the centillion the same way we are continuing the googol. Megafaxul ((200!)!)! The factorial of a kilofaxul. This is about 10^10^10^379. Then we can have a gigafaxul, a terafaxul, a petafaxul, and an exafaxul. 2nd Skewes’ Number e^e^e^e^7.705 ~ 10^10^10^963.5185 Main article: Skewes' numbers This number was proven by Stanley Skewes in 1955 to be an upper-bound to the first crossing point of the prime counting function and the logarithmic integral function, but unlike the original Skewes' number, this bound did not assume the Riemann hypothesis (see my article on Skewes' numbers) to be true. It is bigger than a googolduplex and it held the honor for largest number in a mathematical proof for about 20 years until it was dethroned by Graham's number (actually Little Graham). Googolduplexichime 10^10^10^1000 Or E1000#3. Also a common upper-bound for 2nd Skewes’ Number. Megafuga-five 5^^5 Roughly 10^10^10^10^2184, or E2184#4. A good tetrational number but a tiny pentational number. Googolduplexitoll 10^10^10^10,000 This is equal to ten to the power of ten to the power of a googol to the hundredth power. It falls a little under gagfour. Gagfour A(4,4) = 2^^7-3~ 10^10^10^19,728 A(3,3) wiith the Ackermann function isn't super-impressive at 61, but increase both arguments by one in the function and the value explodes! A(4,4), with the Ackermann function formulas in the entry for 61, can be expressed in up-arrow notation as 2^^7-3 = 22^2^65,536-3 - the vallue falls between 5^^5 and a googolduplexigong. The number is a hugely unfathomable number that passes a googolduplex, and is on the high end of the class 4 range. But gagfive will make this look tiny! 10^(3*10^(3*10^21,000)) Thomas Jones, on his large numbers site, devise a system to name hypercubes of large numbers of dimensions. This number appears to be the limit of the system (note that it can name over a googolduplex dimensions!). A hypercube of this many dimensions, in Jones' system, is called an unyoctillectilletteract. Googolduplexigong 10^10^10^100,000 Continuing Sbiis Saibian’s primitive googol series. The Hyperclass Range10^10^10^1,000,000 ~ 10^^10Entries: 28 Milliontriplex 10^10^10^1,000,000 The boundary between class 4 and class 5 numbers - class 5 numbers range from 10^10^10^1,000,000 to 10^10^10^10^1,000,000. A googoltriplex is an example of a class 5 number. Class 5 numbers are at a stage where x^2 is indistinguishable from x. To get an idea of what that means, imagine a sphere containing a googoltriplex spheres, and then imagine a second sphere made of a googoltriplex of those spheres! Amazingly, the number of particles in the first sphere wouldn't be distinguishable from the number of particles in the second! It's all about the counter-intuition here that really boggles the mind. Pentalogue 10^10^10^10^10 = 10^^5 Or 10^10^10^10^10, fifth member of the -logue series. This number is almost too big to represent anything in science at all. Poincare Recurrence Time (high) 10^10^10^10^13 This is the Poincare recurrence time for a black hole with the mass of the entire universe assuming a certain inflation theory. It's notable because it's arguably the largest number that has any kind of real-world meaning. I say "arguably" because it's uncertain how meaningful this figure really is (it may well be just something hypothetical). For details read this article. Nonetheless this is a truly huge number, far far bigger than anything we can comprehend. After this point there's no number that is known to even arguably represent anything in the real world that can compare to the numbers that lie ahead ... Googoltriplex 10^10^10^10^100 One followed by a googolduplex zeros is usually known as a googoltriplex (pronounced /goo-gol-try-plex/). I've also seen it referred to as "googolplexianite" on analogy to "googolplexian", or "gargantugoogolplex" on analogy to the erroneous "gargoogolplex" for 1 followed by a googolplex zeros. A googoltriplex is the first member of the googol series (googol, googolplex, googolduplex ... ) that cannot be expressed in terms of Bowers' illions. Fugagoogolplex (10^10^100) vv (10^10^100) in down-arrow notation ~ 10^10^10^(10^100+100) This is an awesome number. It looks only somewhat larger than a googoltriplex, but the number of digits in this number is really the number of digits in a googolduplex RAISED TO THE GOOGOLTH POWER! For more see my article on googological prefixes and suffixes. 17^10^(3*10^(3*10^21,000)) The seventeenth unyoctillectilletteract (see 10^(3*10^(3*10^21,000)) ) Pentafact 5* = 6561^^5 ~ E25,043#4 A pentafact is the SpongeTechX mixed factorial of 5, equal to (((1+2)*3)^4)^^5 (see also 6561). It is a very large number that falls between a googoltriplexitoll and 6^^6. Megafuga-six 6^^6 ~ E36,305#4 This is the superfactorial of three; the superfactorial of x is equal to (x!)^^(x!). The function achieves super-tetrational growth rates. Millionquadriplex E6#5 = 10^10^10^10^1,000,000 The boundary between class 5 and class 6 numbers. Class 6 numbers are at a scale where x^log(x) is indistinguishable from than x. For example, imagine a googolquadriplex times a googolquadriplex times a googolquadriplex times a googolquadriplex ... ... ... times a googolquadriplex times a googolquadriplex, where googolquadriplex is said a googoltriplex times. That number is not computably diffrerent from a googolquadriplex itself! Mind-boggling much? Well, hold on to your seats, because soon we'll blast to numbers that transcend any such analogy! 10^10^10^10^18,705,353 A lower bound for BB(7) found by Wythagoras and Cloudy176 of Googology Wiki, which is on the low end of the class 6 range. It's much larger than a millionquadriplex but vanishingly smaller than a hexalogue. Hexalogue 10^10^10^10^10^10 = 10^^6 The sixth member of the -logue series. Joycian Tetratri g(3,3,3,3) A number I defined with Joyce’s g function. It’s approximately E14#5, placing it between a hexalogue and a googolquadriplex. It can be written in up-arrow notation as ((3^^3)^^3)^^3. The g-function has very strange behavior, seen in one of my articles. Googolquadriplex E100#5 A googolquadriplex is one followed by a googoltriplex zeros. It's sometimes referred to as googolquadraplex. Megafuga-seven 7^^7, ~E695,974#5 Falls between a googolquadriplexigong (not listed) and a millionquintiplex. Millionquintiplex E6#6 The boundary between class 6 and class 7 numbers. Heptalogue 10^^7 Googolquintiplex E100#6 One followed by a googolquadriplex zeros is called a googolquintiplex, also called googolquinplex. Millionsextiplex E6#7 The boundary between class 7 and class 8 numbers. Megafuga-eight 8^^8, ~E7#7 Octalogue 10^^8 Googolsextiplex E100#7 Millionseptiplex E6#8 The boundary between class 8 and class 9 numbers. Megafuga-nine 9^^9, ~E9#9 Ennalogue 10^^9 Bentley's Number 10^^9 + 10^^8 + 10^^7 + 10^^6 + 10^^5 + 10^^4 + 10^10^10 + 10^10 + 10 + 1 Bentley's Number is a number based on the story "Forever Endeavor" written by Jonathan Bowers (the name comes from the blog "A googol is a tiny dot"). The story is about an ordinary man named Jack Bentley who takes an opportunity to earn a million dollars by building ten "counters", but the task turns out to be an extremely bad idea, taking virtually eternities upon eternities to finish - the first counter has 1 wheel, the second 10 wheels, the third 10^10 wheels, the fourth 10^10^10 wheels, all the way up to 10^^9 wheels on the last one, and each is used to count the number of wheels on the previous. Bentley's Number is the total number of wheels the main character, named Bentley, has to place on all the counters, and it is the sum of the numbers of the form 10^^n from n = 0 to 9. Googolseptiplex E100#8 Millionoctiplex E6#9 The boundary between class 9 and class 10 numbers. At this point the idea of classes is nearing its breaking point in that it's starting not to become so useful for categorizing numbers. Tetraplo ekatommyriaplekatomyriakis ekatommyrio f(g(f(g(f(g(f(6,000,000))))))) where f(x) = 103x+3 and g(x) = 106x ~ E6,000,000#8 This is a ridiculously huge number that has an interesting story: A man named Harry Foundalis, on his website, has a page that talks about numbers in modern Greek, up to the ridiculously huge 1 followed by three-trillion-and-three zeros (called "picillion" by Jonathan Bowers). He then gives a link to another page which is in Greek about how to name numbers that are truly dizzyingly gigantic, making Bowers' illions look adorable. The system is basically a Greek version of the -illions, which start off like so (phonetically transliterated from the Greek alphabet to the Latin alphabet): million = ekatommyrio billion = disekatommyrio trillion = trisekatommyrio quadrillion = tetrakis ekatommyrio quintillion = pentakis ekatommyrio sextillion = exakis ekatommyrio septillion = eptakis ekatommyrio The names can be thought of as meaning "million", "second million", "third million", etc, since they match with the Greek number roots (e.g. tetra- means 4). Foundalis decides to stretch this system to outrageous heights. The largest number he names with exponent notation is: 6,000,000 3*10 +3 6*10 3*10 +3 6*10 3*10 +3 6*10 3*10 +3 10 O_o;; That monstrous number is named "tetraplo ekatomyriaplekatomyriakis ekatommyria". However it's not the largest number he names! (see ekatommyriaplo ekatommyriaplekatommyriakis ekatommyriaplo ekatommyriaplekatommyriakis ekatommyriaplo ekatommyriaplekatommyriakis ekatommyrio, later) The Tetronomical Range 10^^10 ~ 10^^10^100 Entries: 52 Decker / dekalogue / megafuga-ten 10^^10 This is a personal favorite large number of mine. It’s 10^10^10^10^10^10^10^10^10^10 in full, a power tower of ten tens or ten tetrated to the hundredth. The term dekalogue is from Sbiis Saibian; the term decker is from Bowers. It’s 10^^10 or 10^^^2 in up-arrow notation (a good tetrational number but a tiny pentational number), and E1#10 in Hyper-E. I like this number mainly because it feels like a transition between the smaller numbers and the bigger numbers. The smaller numbers, in my imagination, represent things on Earth. The bigger numbers, however, represent mysterious godly things. This was originally the boundary between part 1 and part 2, but since part 1 was getting too big, I had to change the boundary to a millionduplex, then to a googol, then to a million. By the way, the name "tetronomical" for this range comes from the word "tetronomical" coined by Sbiis Saibian as a portmanteau of "tetration" + "astronomical". Basically exponentiation is to astronomical numbers what tetration is to tetronomical numbers. Googoloctiplex E100#9 A lower bound to the major googolplex and to Genu's Number. Major googolplex (10^10^100)^^8 Used in the definition of Genu's Number. Genu's Number ((1010^100)^^8)((10^3003)^(10^3003)+10^100)+(1010^10^10^10^100)10^6000^10^273 This number was coined in a deleted Googology Wiki blog post by someone who goes by the name Commando Conceptor L5 5.12.159.141 on Googology Wiki. He wrote it as: major googolplexmillillion^millillion+googol+googolquadriplex10^6000^nonagintillion. The creator of this number says that it is the health of the B.L.O.O.N.S.T.O.W.E.R.D.E.F.E.N.S.E.9.W.I.K.I., a fanmade Bloons Tower Defense boss. However, Genu's number is really just a salad number - all the operations performed on the major googolplex have little effect on its size, and it ends up barely any larger than a major googolplex. E101#9 An upper bound to Genu's number more accurate than E109#9. For proof that it's less than E101#9 click here. E109#9 On the page where Genu's Number is introduced to Googology Wiki, Sbiis Saibian proved in a comment that E109#9 is a generous upper bound to Genu's Number. He single-handedly showed that this number is vastly smaller than a googolnoniplex, or E100#10. Googolnoniplex E100#10 A very weak upper bound to Genu's number. Googoldeciplex E100#11 The 10th -plex of a googol. Icosalogue 10^^20 Megafuga-twenty-four 24^^24 ~ E33#23 The superfactorial of four, a power tower of 24 24's. It can also be written as 24^^^2.Penantalogue 10^^50 Giggol / hectalogue 10^^100 A giggol is defined by Jonathan Bowers as equal to ten tetrated to the hundredth, equal to 10^10^10^10^ ... ^10 with 100 tens (a power tower of 100 tens). Written out as a power tower it's: It's a number very analogous to the googol, and it's the first of Bowers' googol extensions. It's one of Bowers' smallest googolisms (can be expressed using array notation as {10,100,2}, since a^..(n ^s)..^b = {a,b,n}), and it was formerly the largest number on Robert Munafo's number list (his current largest listed number is Steinhaus's mega, though he discusses much larger numbers on the large numbers section of his website). Sbiis Saibian has named this number a "hectalogue" from the Greek prefix hecta- meaning 100. Saibian has coined a “similarly sized” number called a grangol, or E100#100. We’ll see it when we get there, but I want to throw in numbers in between to emphasize the two numbers’ real difference, like Saibian did in his number list. Giggol raised to the hundredth (10^^100)^100 A grangol is 10^10^10^10^10^.........10^100 with 100 10’s. It looks like giggol raised to the hundredth but is really a lot larger. Remember the Skewes’ spheres? This number is an impressive-sounding 100th-order giggol sphere, but really not much of an improvement, and much less than a grangol. We’re moving a lot faster. Coogol 10^100 in hypermathematics This is an unusual googolism, equal to 10^100 in hypermathematics. It was coined in the blog "A googol is a tiny dot". Hypermathematics is like normal mathematics, except addition is different: for example, 5+6 = 56, and 30+546 = 30,546. Here’s a visual representation of the coogol: 10 -> 1010...1010 -> 101010.......101010 -> 101010.......101010 -> ... ... ... .... -> 101010.......101010 with 100 stages, where the number of 10s in each stage is the number of the previous stage. This number is between E19#99 and E20#99 in Hyper-E, the former of which is 10^10^ ... ^10^19 with 99 10's and the latter of which is 10^10^ ... ^10^19 with 99 10's. The lower bound is easier to see since you can imagine a coogol as: stage 1 = 10 stage 2 (10101010101010101010) > 10^19 (20 digits long) stage 3 (1010101.......010 with stage 2 10's) (2*stage 2 digits long) > 10^stage 2 = 10^10^19 = E19#2 stage 4 (1010101.......010 with stage 3 10's) (2*stage 3 digits long) > 10^stage 3 = 10^10^19 = E19#2 stage 5, with this pattern, ~ E19#4 stage 6 ~ E19#5 ........ coogol = stage 100 ~ E19#99, placing it between giggol and megafuga-hundred, still well under grangol. Megafuga-hundred 100^^100 This number is greater than a giggol but not 10^^101, let alone a grangol. It’s between E200#99 and E201#99. 10^^101 This number is what we get when we do 1 followed by a giggol zeroes. As you can see 1 followed by x zeroes is getting obsolete with numbers this large - this value is still under a grangol. Giggolbang (10^^100)! With numbers this big, x! is between 10^x and x^x; therefore this number falls just within the cracks of 10^^101 and fzgiggol. Fzgiggol (10^^100)^(10^^100) Remember the 100th-order giggol sphere? This is a giggolth-order giggol sphere! As incredible as this sounds, it’s STILL less than a grangol! Even if you try raising THAT to the giggolth power, it still doesn't get us far at all, not even past a grangol. E11#100 An upper bound to a fzgiggol. f3(100) ~ E32#100 This is another example of a number definable in the fast-growing hierarchy. It's a mind-blowingly large number far larger than any number to appear in even the most exotic realms of science! The value is larger than a giggol but falls short of a grangol. Grangol E100#100 A grangol is one of Sbiis Saibian's smallest googolisms, analogous to the googol and comparable to the giggol (although it's larger than raising a giggol to its own power). It is also equal to 10^10^10 ... ... ... ^10^100 with 100 tens, a power tower of 100 10s topped off with 100. Sbiis Saibian said that it is The grangol can be thought of as googol-99-plex. It also be written in Andre Joyce's g-function as g(101,1,2,2,10). 10^^102 The smallest power tower of 10s more than a grangol. Fugagiggol (10^^100) vv (10^^100) This can also be expressed as (10^^100)^(10^^100)^(10^^100-1). It's barely smaller than the next number, a giggol tetrated to the third, but also by necessity larger than the previous number. Here's why: (10^^100)^(10^^100)^(10^^100-1) > 10^(10^^100)^(10^^100-1) = 10^(10^(10^^99*(10^^100-1))) > 10^(10^(10^^100)) = 10^(10^^101) = 10^^102 (10^^100)^^3 This is a giggol tetrated to the third. It is bigger than a grangol, but not that much (at least in terms of power tower height of course). In fact, it’s only the 10th power of a fugagiggol, trivial compared to even the difference between a giggol and grangol. Googolcentiplex/grangolplex E100#101 With Saibian’s numbers, -plex means 10 to the power of a number. This number is used to demonstrate the inadequacy of the -plex suffix on numbers this big; it makes the power tower of 10s only one level high. This is why Sbiis Saibian invents the suffix -dex which has a much bigger effect on the number, and for bigger numbers gives up on a suffix system and uses the word "grand" (grand godgahlah, grand tethrathoth). This is the same value as a googolcentiplex, googol’s 100th -plex, and the two numbers happen to be the same value. Because of that, grangolplex/googolcentiplex is an example of what defines a googolism - I think in general googolism really refers more to a name for a number than a number for a name. Therefore grangolplex and googolcentiplex are probably two different googolisms, and in fact they have two separate articles on Googology Wiki, which further reflects this. Fore more on what defines a googolism see my entry for 21 in part 1 of this list. 10^^103 An upper bound to a grangolplex. 10^^198 A power tower of 10's used as a lower bound to the expofaxul. E5#198 A more accurate lower bound to an expofaxul. E183,230#197 A very accurate lower bound to an expofaxul. Expofaxul 200!1 Another Hollomism, defined with the first part of hyperfactorial array notation. Basically: x!y = x{y}(x-1){y}(x-2){y}........3{y}2{y}1. Remember that it’s evaluated from right to left and that a{c}b is a^^^...^^^b with c ^s. So, this is 200^199^198^197.......^5^4^3^2^1. The power tower is approximately twice as large as that of a giggol. The value falls between 10^^198 and 10^^199. E183,231#197 A very accurate upper bound to an expofaxul. E6#198 A moderately accurate lower bound to an expofaxul. 10^^199 A power tower of 10's used as an upper bound to an expofaxul. 10^^200 This number is notable because it looks like giggol^giggol when written as a power tower, but is in fact much larger. It's also a rough upper bound to an expofaxul. 10^^257 A power tower of tens commonly used as a lower bound for Steinhaus's mega. E619#256 A more accurate lower-bound for the mega. It's one of the easiest 10-based bounds to prove, and written as a power tower it looks like: 10^10^10^ ... ... ^10^10^619 w/256 10s See my article on Steinhaus-Moser notation for a proof of this bound. Steinhaus's Mega 2 in Steinhaus-Moser notation ~ E619.3#256 Main article: Steinhaus-Moser Notation x is x in a triangle, or x^x. x is x in a square, or [...[[[x in a triangle] in a triangle] in a triangle] ... ... in a triangle] with x triangles. x is x in a pentagon, or [...[[[x in a square] in a square] in a square] ... ... in a square] with x squares. and so on. That’s Steinhaus-Moser notation, a notation devised by mathematician Hugo Steinhaus and extended by Leo Moser as an example of how easy it is to name very large numbers - see the link above for details on how the notation came to be. The mega was coined by Steinhaus along with the megiston in his book Mathematical Snapshots, while the Moser was coined by Leo Moser (it is unknown where it was first defined). The mega is one of the “classic” large numbers together with the googol and googolplex, Skewes’ number, the infamous "world's largest number" Graham’s number, and (among others) the megiston and Moser. The last two of those are also defined with Steinhaus-Moser notation. The mega, or 2 in a pentagon, is already a very large number as you can see, which is equal to 2 in a pentagon, which solves to: pentagon(2) = square(square(2)) = square(triangle(triangle(2)) = square(triangle(22)) = square(triangle(4)) = square(44) = square(256) = triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle(triangle( triangle(triangle(triangle(triangle(triangle(triangle(256))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) )))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) ))))))))))))))))))))))))))))))))))))) (there are 256 triangles) We can also denote this triangle256(256), where triangle(n) is n in a triangle and the exponent on the function indicates how many times you apply the triangle function to 256. It's amazing just how big such a simple number is, and yet it isn't too hard to estimate how big it really is (see my article on Steinhaus-Moser notation for how you can estimate it). But it pales in comparison to the megiston, defined as 10 in a pentagon! This is also the largest number on Robert Munafo's number list - he says that it is the last one because soon after this number, numbers become hard to express in any form at all. However, he discusses much larger numbers in the large numbers section of his website. E620#256 A more accurate upper-bound on the mega. It's a little harder to prove than E874#256 but still pretty easy. Written out as a power tower it looks like: 10^10^10^ ... ... ^10^10^620 w/256 10s E874#256 Another upper bound to Steinhaus's mega that is a little easier to prove than E620#256. Written out as a power tower it looks like: 10^10^10^ ... ... ^10^10^874 w/256 10s 10^^258 A power tower of tens used as an upper-bound on the mega. 256^^512 This is a fairly naive upper-bound on Steinhaus's mega, which can be derived using Sbiis Saibian's Knuth Arrow Theorem. Since by the Knuth Arrow Theorem (a^^b)^^c < a^^(b+c) and the mega = triangle256(256) and triangle(n) = n^n = n^^2, we can see that: triangle(256) = 256^^2 triangle2(256) = triangle(triangle(256)) = (256^^2)^^2 < 256^^4 triangle3(256) = triangle(triangle2(256)) < triangle(256^^4) = (256^^4)^^2 < 256^^6 triangle4(256) = triangle(triangle3(256)) < triangle(256^^6) = (256^^6)^^2 < 256^^8 ... triangle256(256) = triangle(triangle255(256)) < triangle(256^^510) = (256^^510)^^2 < 256^^512 As it turns out this is  a much weaker upper-bound than you might think. It's about E619#511 (a power tower of 511 tens topped off with 619), in comparison to the mega which is about E619#256 (a power tower of 256 tens topped off with a 619) - the power tower representing this upper-bound is about twice as tall as the power tower that represents the mega. 10^^514 A pwer tower of 10's that itself upper-bounds the bound 256^^512 for the mega. This bound can be shown since: 256^^512 < 10,000,000,000^^512 = (10^^2)^^512 < 10^^(2+512) = 10^^514 Chilialogue 10^^1000 This is a power tower of 1000 tens. It's a new member of Saibian's -logue family - it comes from Greek "chilias" meaning a thousand. Grangolchime E1000#1000 A power tower of 1,000 tens topped with 1,000. Myrialogue / Joycian gaggol 10^^10,000 A power tower of 10,000 tens. The name comes from Greek "myria" meaning ten thousand. This number is also equal to Andre Joyce’s (former) mistaken definition of the gaggol. The real gaggol is equal to 10^^^100 = 10^^10^^10^^10......^^10 with 100 10s, which is unfathomably larger. The weird thing there is the Joyce's g function can represent the correct value of a gaggol as g(4,100,10). Grangoltoll E10,000#10,000 A power tower of 10,000 tens topped off with 10,000. Two pentated to four / Two hexated to three 2^^^4 / 2^^^^3 ~ E19,729#65,532 This is a cool number which can be expressed in several different ways using Knuth's up-arrow notation. Let’s start by simplifying the number from the form 2^^^^3: 2^^^^3 = 2^^^2^^^2 = 2^^^2^^2 = 2^^^2^2 = 2^^^4 = 2^^2^^2^^2 = 2^^2^^2^2 = 2^^2^^4 = 2^^2^2^2^2 = 2^^2^2^4 = 2^^2^16 = 2^^65,536 = 2^2^2^2^2.........^2^2 with 65,536 2s So it can be expressed with hexation as 2^^^^3, pentation as 2^^^4, tetration as 2^^65,536, and exponentiation as 2^2^2^2^2.........^2^2 with 65,536 2s. So it’s a very very small hexational number, a small pentational number, but a large tetrational number. Amongst Sbiis Saibian's googolisms it falls between a grangoltoll and a giggolgong. For more on numbers like this read my article on Knuth's up-arrows. Giggolgong 10^^100,000 This is a googolism coined by Sbiis Saibian by combining the -gong suffix with Bowers' giggol. It's a power tower of 100,000 tens, a thousand times taller than the tower used to represent a giggol. Grangolgong E100,000#100,000 A power tower of 100,000 tens topped with a 100,000. The -gong suffix regularly appears in Saibian’s smaller numbers. Megafuga-million 1,000,000^^1,000,000 This number serves as an example of just how many numbers can be named with the megafuga- prefix. It's a power tower of a million millions. Joycian gygol 10^^10,000,000 Andre Joyce also attempted to define the Bowerian gigol (at the time called gygol). The real gigol is equal to 10^^^^^100, which makes the Joycian gygol look tiny. Joyce’s g function can represent the gigol as g(6,100,10). How did he not know that? Octadialogue 10^^100,000,000 An ocatadialogue is defined by Sbiis Saibian as equal to ten tetrated to the one-hundred-millionth, a.k.a. a power tower of 100 million tens. Grangolbong E100,000,000#100,000,000 A power tower of 100 million tens topped off with 100,000,000. It's called grangolmine by Aarex. Output of pete-9.c ~ 2^^386,201,107 pete-9.c was an entry in Bignum Bakeoff by the guy named Pete - it was supposed to improve on pete-8.c, which was already a lot WORSE than his other programs, but ended up making a number that was a tiny bit smaller (see next entry). Its only difference from pete-8.c was that it used 99 in one part instead of 999.Bignum Bakeoff entries:< pete-3 | pete-9 | pete-8 > Output of pete-8.c ~ 2^^386,201,107 pete-8.c was an entry in Bignum Bakeoff by the guy named Pete, which was meant to improve the quite powerful pete-7.c. However, a bug caused the program to be far weaker, making a number that's FAR FAR SMALLER than the output of pete-7.c.Bignum Bakeoff entries:< pete-9 | pete-8 | harper > Joycian gagol 10^^1,000,000,000 Joyce attempted to define the Bowerian gagol as well. The real gagol is equal to 10^^^^^^^100 (that’s 7 ^s). In Joyce’s g function, the gagol is equal to g(8,100,10). Perl Hypercalc limit 10^^10^10 This number is approximately the limit of the Perl version of Robert Munafo's Hypercalc, an awesome calculator program useful for computing numbers up to the tetronomical level (which is where we are right now). This limit is ABSURDLY larger than most calculators' limits, often 10^100 or 10^1000. Grangolthrong E100,000,000,000#100,000,000,000 A power tower of 100 billion tens topped off with 100,000,000,000. Tritri 3^^^3 This staggering number, name coined by Bowers, is called tritri because it’s {3,3,3} (3 threes) in array notation. Here’s how it’s evaluated: 3^^^3 3^^3^^3 3^^3^3^3 3^^3^27 3^^7,625,597,484,987 3^3^3^3........^3 with 7,625,597,484,987 3’s, an unfathomable sized number. The best way to get an idea of how big tritri is is by considering the power-tower of threes: imagine the power-tower of those 7 trillion threes reached from Earth to the sun - each three would be about 2 centimeters in size. Then, the topmost five threes would already form a number undescribably larger than a googolplex, and that's only the top 10 centimeters of the tower - imagine how big the whole number must be if we can't comprehend the tip of the tower!!! This is one of only two Bowersisms Joyce gets right when he defines them - the other is the tridecal. Tritri is notable as part of a moral thought experiment in the group blog LessWrong: "Would you prefer that one person be horribly tortured for fifty years without hope or rest, or that 3^^^3 people get dust specks in their eyes?" Of course, many thinking of the question may get lost in trying to contemplate tritri's size - it's a nice way to boggle the shit out of your mind, and bring home its size (or more, bring home how you CAN'T bring home its size). In my opinion, it's a good example of an unfathomable number, largely since it's pretty easy to explain. It's one of my favorite numbers larger than a googolplex. Sedeniadalogue 10^^(10^16) A sedeniadalogue is defined by Sbiis Saibian as equal to a power of ten quadrillion tens. That means that it is far larger than tritri but EASILY beaten by the googol-stack or even the guppylogue. Guppylogue 10^^(10^20) Sbiis Saibian continues the -logue idea by appending some of his own googolisms with the -logue suffix; this number and the following few numbers nicely fill the gap between a sedeniadialogue and a googol-stack. Minnowlogue 10^^(10^25) Gobylogue 10^^(10^35) Gogologue 10^^(10^50) 256^^(2^256) ~ 10^^(1.16*10^77) This is a very weak upper-bound on the mega. It is can be shown by considering: triangle(256) = 256^256 triangle2(256) = (256^256)^(256^256) triangle3(256) = ((256^256)^(256^256))^((256^256)^(256^256)) triangle4(256) = (((256^256)^(256^256))^((256^256)^(256^256)))^(((256^256)^(256^256))^((256^256)^(256^256))) etc. Since the expression for triangle(256) has 2 256's, the expression for triangle2(256) has 4 256's and is less than 256^256^256^256 = 256^^4, triangle3(256) has 8 256's and is less than 256^^8, etc, we can continue this to get triangle256(256) < 256^^(2^256) ~ 10^^(1.16*10^77). Even this upper-bound is vastly smaller than the megiston, which is 10 in a pentagon in contrast with mega which is just 2 in a pentagon. It doesn't even make it to the size of a googol-stack since the power tower of 10's representing a googol-stack is about 100 sextillion times taller than the tower representing this number. Ogologue 10^^(10^80) The Hyper-Tetronomical Range 10^^10^100 ~ 10^^10^^100 Entries: 25 Googol-stack / Googologue / Googolgoogolplex 10^^(10^100) This is a power tower of a googol tens, or ten tetrated to a googol. It comes from the Cantor's Attic suffix -stack which turns n into 10^^n. Sbiis Saibian calls this number "googologue" using his -logue suffix. Additionally the Big Psi web book on large numbers calls this a "googolgoogolplex". Googoldex E100#1#2 Here’s how the -dex suffix works: If x = Ea#b#c, then x-dex = Ea#b#(c+1). A googol can be written as E100#1#1, so this is a googoldex. E100#1#2 = E100#(E100#1#1) = E100#(E100) = E100#googol. So this is a power tower of 10s a googol terms high topped off with a 10. It falls between googol-stack and megafugagoogol. Megafugagoogol (10^100)^^(10^100) A megafugagoogol is a number equal to a power tower of a googol googols, an example of a number nameable with the megafuga- prefix. It is only slightly larger than the googoldex - to see why consider this: (10^100)^^(10^100) > E(10^100)#(10^100-1), a power-tower of a googol-minus-one tens topped off with a googol, which is equal to E100#(10^100) = googoldex See also megafugagoogolplex. Ecetondex E303#1#2 Extending the centillion like the googol. This is a power tower of a centillion 10s, topped off with a 303, and an example of the kind of number Sbiis Saibian came up with as a kid. Limit of Javascript Hypercalc 10^^(1.7976*10308) The Javascript version of Hypercalc is even more wide-ranged than the Perl version, maxing out at a number around the level of a power tower of 1.7976*10308 tens. That makes it technically wide-ranged enough to store numbers like tritri or a googoldex! However, the Perl version has advantages of being fully programmable and additional features. Googoldexigong E100,000#1#2 Just for the hell of it, here's the "gong" version of the googoldex - a power tower of a googolgong 10s, topped off with a 100,000. Zootzootplex googolplex!1 This number is googolplex^(googolplex-1)^(googolplex-2).....^5^4^3^2^1, and was coined by some guy named Andrew Schilling at the age of four (according to a source, he seems to be about a year older than me). It's equal to the expofactorial of a googolplex. Schilling also made up two humorous zillion-like names: sillion is "zillion, but used in sillier contexts", and sillyillion is "like sillion but much larger, like millillion or milli-millillion". Since the zootzootplex is a power of a googolplex it’s a huge power of 10. With Hypercalc we find that when x≥6, x expofactorial is about E(183,230.683)#(x-4). Therefore this number is comparable to a power tower of a googolplex minus 4 tens topped with a 183,230, and therefore it's upper-bounded by that power tower topped off with 10,000,000,000, a.k.a. 10^^(googolplex-2) or a power tower of a googolplex minus 2 tens. Zootzootplex, with these calculations, therefore can be estimated to fall under both googolplex-stack and googolplexidex. Googolplex-stack 10^^(10^10^100) You can combine number suffixes to create more complicated googolisms, such as this number, a power tower of a googolplex tens. This number is also a lower bound for the googolplexidex. Googolplexidex E100#2#2 A power tower of a googolplex 10s topped off with a 100. Upside-down zootzootplex 2^3^4^5^6^.......^(googolplex-2)^(googolplex-1)^googolplex The webpage where the zootzootplex came from says that the zootzootplex would be larger if it were flipped on its head - that is, the power tower is reversed (starting with a 2 so that it doesn't degenerate into 1). The number is comparable to and just barely bigger than a power tower of a googolplex tens topped off with a 100, making it just barely bigger than a googolplexidex. Googolplexidexiplex E100#(10^10^100+1) This is an example of a number that can be named by combining googological affixes: in this case, using plex, then dex, then plex again. It's equal to a power tower of a googolplex plus one tens topped with a 100. It's also a lower bound for the megafugagoogolplex. E(100+10^100)#(10^10^100) This number is a relatively tight lower bound for the megafugagoogolplex, proven by Sbiis Saibian. Megafugagoogolplex (10^10^100)^^(10^10^100) This number is a power tower of a googolplex googolplexes, nameable with the megafuga- prefix. It's an insanely huge number, though that's mostly because of the height of the power tower, not as much because every number in the tower is a googolplex. It is an example of a large number a non-googologist might come up with, often to try to trump Graham's number (it doesn't even come close!). To see why we'll use Sbiis Saibian's Knuth Arrow Theorem, which states that for a≥2, b≥1, c≥1, x≥2, (a^xb)^xc < a^x(b+c): (10^10^100)^^(10^10^100) < (10^10^10^10)^^(10^10^100) = (10^^4)^^(10^10^100) < 10^^(4+10^10^100) < 27^^(4+10^10^100) = (3^^2)^^(4+10^10^100) < 3^^(6+10^10^100) < 3^^(10^10^101) < 3^^(10^10^10^10) = 3^^(10^^4) < 3^^(27^^4) < 3^^((3^^2)^^4) < 3^^(3^^6) < 3^^3^^3^^3 = 3^^^4 << 3^^^3^^^3 = 3^^^^3 = G1 <<< G64 = Graham's number Sbiis Saibian proved that megafugagoogolplex is between E(100+10^100)#(10^10^100) and E(101+10^100)#(10^10^100), and therefore it falls between a googolplexidexiplex and a googolplexidexiduplex. E(101+10^100)#(10^10^100) Sbiis Saibian proved this number to be a relatively strong upper bound to the megafugagoogolplex. Googolplexidexiduplex E100#(10^10^100+2) A power tower of a googolplex plus 2 tens topped off with a 100. This number is a weaker upper bound to the megafugagoogolplex. Output of harper.c ~ 2^^(2^2^100,000,002), about 10^^(10^10^30,102,999) This number is the output of harper.c, Russell Harper's entry in Bignum Bakeoff. His program performs iterated exponentiation to produce a very large tetrational number - a number comparable to a power tower of about 10^10^30,000,000 twos that falls somewhere between a googolplexidex and a googolduplexidex. David Moews, the man who held Bignum Bakeoff, comments that Harper's program was inefficiently written; he showed that the program can be rewritten to produce the same number, but only use half as many characters. Googolduplexidex E100#3#2 A power tower of a googolduplex tens topped off with 100. Googolplex-bang-stack 10^^((10^10^100)!) This number is a power tower of 10s about a googolduplex terms high, so it’s about E100#3#2. It’s a bit more than a googolduplexidex. Googolbang-plex-stack 10^^(10^((10^100)!)) The largest number formable by combining the googol with one each of -bang, -plex, and -stack. Googoltriplexidex E100#4#2A power tower of a googoltriplex tens topped off with 100.Megafugagargantugoogolplex(10^10^10^10^100)^^(10^10^10^10^100)A megafugagargantugoogolplex is a googoltriplex tetrated to a googoltriplex. Its name comes from "megafuga-" plus "gargantugooogolplex", an alternate name for a googoltriplex coined on analogy to the erroneous "gargoogolplex" for 1 followed by a googolplex zeroes (it's actually a googolplex squared). It's the largest number listed on a few online large number lists, and therefore it's sometimes touted as an incredibly huge number; I think this is also partly because its name has an appeal of sounding really long and mighty, prepending the googolplex with a bunch of catchy-sounding roots. One random user on Googology Wiki erroneously claimed it to be the largest named number (it obviously isn't). Joycian pentatri g(3,3,3,3,3) Another number I defined with Joyce’s g function. It’s approximately E14#5#2, placing it between a googoltriplexidex and a googolquadriplexidex. Googolquadriplexidex E100#5#2 Tria-taxis / deckerplex 10^^^3 A power tower of 10s a decker tens high. X-taxis (term by Saibian, formerly -teraksys) is 10^^^x. I have seen this number called "deckerplex" based on Bowers' name "decker" for 10^^10. The Pentation Range 10^^10^^100 ~ 10^^^(10^100) Entries: 39 Giggolplex 10^^10^^100 Unlike Saibian, Bowers allows -plex to perform any type of recursion. This number is a power tower of a giggol 10s, and is the first recursive step after a giggol. Grangoldex E100#100#2 This number is a power tower of 10s a grangol terms high, topped off with a 100. It's comparable to giggolplex. Kiloexpofaxul expofaxul!1 The exponential factorial of an expofaxul, or expofaxul^expofaxul-1....^4*3*2*1. It should be obvious what a megaexpofaxul, gigaexpofaxul, etc. are. Grangoldexigong E100,000#100,000#2 Yet another -gong number; this is a power tower of a grangolgong 10's topped with a 100,000. Triton / Grand Mega 3 in Steinhaus-Moser Notation A triton or grand mega is 3 in a circle or pentagon in Steinhaus-Moser Notation. The first name was coined by Sbiis Saibian (source: personal communication), and the second by Aarex on his large number site. It falls between 3^^^4 and 3^^^5 in up-arrow notation, and it can be estimated like so: Square(x) is in general roughly 10^^x Pentagon(3) = square(square(square(3))) = square(square(triangle(triangle(triangle(27))))) = square(square(triangle(triangle(27^27)))) ~ square(square(triangle(10^10^40)))) ~ square(square(10^10^10^40) ~ 10^^10^^10^10^40, between a giggolplex and a giggolduplex 2^^2^^2^^9 This value is the current upper bound to the problem upper-bounded by Graham's number, proven in 2013. It is vastly smaller than Little Graham, much smaller than G(2) or even G(1). It falls between a grand mega and a tetra-taxis. The current lower bound is 13. Tetra-taxis 10^^^4 Taxis was chosen to indicate that the numbers are tetrational towers. Giggolduplex 10^^10^^10^^100 A power tower of a giggolplex 10s - the second recursive step from a giggol. Grangoldudex E100#100#3 A power tower of a grangoldex 10s topped off with a 100. Ekatommyriaplo ekatommyriaplekatommyriakis ekatommyriaplo ekatommyriaplekatommyriakis ekatommyriaplo ekatommyriaplekatommyriakis ekatommyrio E(E(E(106))) This is the largest number Harry Foundalis names in his extended system to the Greek numbers' names. It exhausts exponential notation entirely and is defined with a function E(x) defined as: E(1) = 103*10^6,000,000+3 x > 1: E(x) = 103*10^(6*E(x-1))+3 E(x) can be estimated to be equal to about 10^^(2x+1) in terms of Knuth's up-arrows, or more crudely 10^^x when working with larger inputs. Therefore the entire number can be estimated as: 10^^10^^10^^2,000,001 YIKES!!! That's a RIDICULOUSULY huge number named with a system, and it UTTERLY leaves even Bowers' illions in the dust! It's a pentational number which would laugh at ANY of Bowers' illions. f4(4) ~ E21#3#4 or 10^^10^^10^^10^10^10^21 This number is equal to f4(4) or fw(4) in the fast-growing hierarchy - it's an ABSOLUTE MONSTER OF A NUMBER!! It leaves all the weeny tetrational numbers like f3(100) in the dust and is a decent pentational number not too much less (relatively speaking) than 5^^^5. Great Mega 4 This number (once again by Aarex) is about 10^^10^^10^^10^10^10^617, or more crudely 10^^10^^10^^10^^4, so it’s between f4(4) and boogafive. Boogafive 5^^^5 5^^^5 is the next member of the booga- sequence after 4^^4 - the value is monstrous and can be imagined like so: 5^5^5^5^5^5^5........^5 w/ 5^5^5^5^5^5^5........^5 5s w/ 5^5^5^5^5^5^5........^5 5s w/ 5^5^5^5^5 5s The value falls between a great mega and a penta-taxis. Penta-taxis 10^^^5 Giggoltriplex 10^^10^^10^^10^^100 Gong Mega 5 This is 5 in a pentagon with Steinhaus-Moser notation, a number that I've estimated falls between a giggoltriplex and a hexa-taxis. Hexa-taxis 10^^^6 Hexomega 6 After that we have a heptomega, octomega, and a nonomega (which I prefer calling an ennomega). Gagfive A(5,5) = 2^^^8-3 ~ E4#65,533#5 This is the next member of the gag- series - its computation is very hard to visualize and it falls between a hexa-taxis and a hepta-taxis. Hepta-taxis 10^^^7 Octa-taxis 10^^^8 Ena-taxis 10^^^9 Deka-taxis/Gigafuga-ten 10^^^10 Gigafuga-x (a prefix I made) is x^^^x. It's mostly just for fun. Endeka-taxis 10^^^11 A lower bound for the megiston. Megiston 10 Main article: Steinhaus-Moser Notation This is the second of the three official Steinhaus-Moser numbers, defined by Steinhaus as 10 in a pentagon in Steinhaus-Moser notation. It has been mistakenly called megistron and also megaston, and for a while Googology Wiki called it megistron. A megiston is much much larger than a mega but much much much smaller than Moser. It’s between 10^^^11 and 10^^^12. It's a number that, unlike the mega, takes full advantage of the pentagon operator to produce a decent pentational number! It's also harder to bound than Steinhaus's mega. Here’s how to calculate a megiston: First take 10, take it to the power of itself, take that to the power of itself, then take THAT to the power of itself, and do it 10 times total. The end result is stage 1. Then take stage 1, take it to the power of itself, take that to the power of itself, then take THAT to the power of itself, and do it stage 1 times total. The end result is stage 2. Continue, and this number is stage 10! However, the Moser, which was coined by Moser instead of Steinhaus, just transcends both the mega and megiston! Dodeka-taxis 10^^^12 An upper bound for the megiston. 10^^^68 A lower bound to Genu's Number II. Genu's Number II 70!2*35!2*812,500*812,500812,500 The guy who coined Genu's Number decided to post a sequel to the number on another Wikia wiki. He called it Genu's Number II in an attempt to trounce his previous number. Though it's unfathomably larger than the previous number, unfortunately it's still a salad number ... a really really bad one. It can be intuitively recognized as such, and in fact it can be shown that: 10^^^68 << Genu's Number II << 10^^^71 (for proof of this click here). This also shows that Genu's number is much less than a gaggol. 10^^^71 An upper bound to Genu's Number II. Gaggol / hecta-taxis 10^^^100 A gaggol (googolism by Bowers) is equal to ten pentated to the hundredth, or {10,100,3} in array form. It is another googolism analogous to the googol and giggol. Here is how to imagine gaggol: Stage 1 = 10 Stage 2 = 10^10^10^10^10^10^10^10^10^10 (10 10’s) = decker Stage 3 = 10^10^10............^10^10 with stage 2 10’s Stage 4 = 10^10^10............^10^10 with stage 3 10’s .... Stage 100 = gaggol. Gigafuga-hundred 100^^^100 Sbiis Saibian says that this number might be considered "the largest pentational number" if you consider 100 to be the largest smallish argument in a pentation expression. He himself notes that this is an arbitrary designation. Megafuga-gaggol (10^^^100)^^(10^^^100) Make a power tower of a gaggol gaggols, and you still get something short of a greagol. Improvements that sound amazingly mind-boggling are often really rather trivial. Greagol E100#100#100 A greagol (great googol) is the next major Saibianism. It’s comparable to a gaggol, even though gaggol^^gaggol3->2->2 in chained arrow notation, which simplifies to 3->3->(3->3->1->2)->1 = 3->3->(3->3) = 3->3->27 = 3^^^...(27 ^s)...^^^3. Boogahundred 100{98}100 Notable for being "a little" less than a boogol. Boogol 10{100}10 or {10,10,100} A boogol is a frighteningly large Bowerian googolism, equal to 10 "dohectated" (the 102nd hyperoperator) to the hundredth. It expands to: 10^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^10 and it's yet another number analogous to the googol. Gugold E100##100 This Saibianism (short for golden googol) is comparable to a boogol, and comparable to 10 102-ated to the 100th. It’s the end of Hyper-E and the beginning of Extended Hyper-E. The main rules of Extended Hyper-E mimic up-arrow notation: Ex##y=Ex#x#x#x#x......#x#x with y x’s Ex###y=Ex##x##x##x##x......##x##x with y x’s etc. So this number can be written in regular Hyper-E as: E100#100#100#100...................#100 with 100 100’s. Hyperfaxul 200! = 200!200 This is 200{200}199{200}198{200}......4{200}3{200}2{200}1, making it broadly comparable to Bowers' boogol. It's expressible as 200! using the very beginning of Hollom's hyperfactorial arrays since x! = x!x. Gamoogol g(1000,100,10) = 10{999}100 The continuation from a gaxoogol. It doesn't seem quite completely right, but I guess it works. Gamoogood g(1000,1000,1000) = 1000{999}1000 An odd jump from the previous number - it's kind of confusing, and hard to see where Joyce will go from here. Gamamoogood g(1000,1000,1000,1000) This is another odd continuation. It can be imagined as: a1 = 1000{999}1000 a2 = a1{999}1000 a3 = a2{999}1000 ..... gamamoogood = a1000 By Sbiis Saibian's Knuth Arrow Theorem, (a{b}c){b}d ~< a{b}(c+d). Therefore, a2 can be upper bounded by 1000{999}2000, a3 can be upper bounded by 1000{999}3000, etc. so this whole number can be upper bounded by 1000{999}1.000.000. Gamamamoogood g(1000,1000,1000,1000,1000) A usage of the 5-entry g-function. It's roughly 1000{1000}1000. making it vastly smaller than the next entry. This is still smaller than the gamamamamoogood, equal to g(1000,1000,1000,1000,1000,1000), seen after Graham's number. Boogolgong 10{100,000}10 “Gonging” another Bowersism, just for the hell of it. Gugoldagong E100,000##100,000 Or E100,000#100,000#100,000#100,000.......#100,000 in Hyper-E with 100,000 100,000s. This number is about 10{100,000}100,001, so it would be comparable to a boogolgong. Gongol 10{10^100 - 2}100 This number, like the coogol, came from the blog "A googol is a tiny dot". It's equal to 10 googolated to the 100th. The number was made just for fun. In a variant of the 3-argument "hyper function" (a multi-argument version of up-arrow notation sometimes used in computer programs), it's equal to hyper(10,10^100,100). Boogagoogol / Med 10^100{10^100-2}10^100 This number was formerly named along with the bed (trooga(googol)) on Googology Wiki - both are among the deleted googologisms because they were made up on the wiki with no sources. Gaggoogol A(10^100,10^100) = 2{10^100-2)10^100+3 This is the gag- prefix applied to the googol, a.k.a. the result of calling the Ackermann function with a googol as both of the arguments. The value is comparable to and slightly larger than the boogagoogol, but smaller than Sbiis Saibian's great googol. 10{10^100}10 This number is probably about the largest up-arrow notation level number, seeing as the googol to me is the limit of "smallish" arguments and the number of up-arrows matters much more than the numbers before or after. Once again, this designation is arbitrary. Saibian's great googol E100##1#2 This is an extended Hyper-E number by Sbiis Saibian, which he calls great googol. It simplifies to E100#100#100#100...............#100 with a googol 100's, and it's about 10{10^100}101 (upper-bounded by 10{10^100}102), making it comparable to a gongol. See also great googondol, equal to E100##10#2. There is also a number Andre Joyce named great googol, equal to exactly 10^1000, which Saibian calls googolchime - see the entry for 10^1000 for more. These numbers may seem huge, but they're just preliminaries to some far more horrifying numbers - welcome to part 4 of this list. Click here for part 4
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