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 ~ f_w(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 made this website is 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.

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 = 2^{2^2^65,536}-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 Range

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) = 10^3x+3 and g(x) = 10^6x

~ 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

((10^{10^100})^^8)^{((10^3003)^(10^3003)+10^100)}+(10^{10^10^10^10^100})^{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 googolplex^{millillion^millillion+googol}+googolquadriplex^{10^6000^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:

As you can see this number's computation is still visualizable. Tritet doesn't have the same charm G(1) has as just barely visualizable since the additional layer of iterated power towers makes it harder to understand the number.

Tetra-petaxis

10^^^^4

Gaggolduplex

10^^^10^^^10^^^100

Greagolduthrex

E100#100#100#3

Penta-petaxis

10^^^^5

Boogasix

6^^^^6

Can be imagined as:

6^6^6....^6 w/ 6^6^6....^6 sixes w/ 6^6^6....^6 sixes ......... w/ 6^6^6....^6 sixes w/ 6 sixes

w/ 6^6^6....^6 w/ 6^6^6....^6 sixes w/ 6^6^6....^6 sixes ......... w/ 6^6^6....^6 sixes w/ 6 sixes power towers

w/ 6^6^6....^6 w/ 6^6^6....^6 sixes w/ 6^6^6....^6 sixes ......... w/ 6^6^6....^6 sixes w/ 6 sixes power towers

w/ 6^6^6....^6 w/ 6^6^6....^6 sixes w/ 6^6^6....^6 sixes ......... w/ 6^6^6....^6 sixes w/ 6 sixes power towers

w/ 6^6^6....^6 w/ 6^6^6....^6 sixes w/ 6^6^6....^6 sixes ......... w/ 6^6^6....^6 sixes w/ 6 sixes power towers

w/ 6 power towers

YIKES! The value is by now iterating the degree of iteration of power towers - by now it's very difficult to wrap your mind around.

Deka-petaxys / terafuga-ten

10^^^^10

Terafuga-x (I made the prefix) is x^^^^x.

A-oomega

10[6]

Or 10 in a hexagon. The is equal to megiston with the powerful pentagon operator applied to it, NINE TIMES!

Geegol / hecta-petaxis / gavoogol

{10,100,4} = 10^^^^100 = g(5,100,10)

Yet another Bowerian googolism analogous to the googol, equal to ten hexated to 100. This number can be imagined as:

Stage 1 = 10 [going back to the stages for gaggol]

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

etc.

Geegol is stage stage stage ... ... stage 1 with 100 stages. A dizzying value, but still not that big a googolism.

Andre Joyce named this number gavoogol, since ga-x-oogol is apparently g(x,100,10) (which is equal to 10^^^...^^^100 with x-1 ^s). It's the first member in the family of numbers formed by Andre Joyce when he supposedly "got high", which I call the gamoogol family. The next is gaxoogol, equal to g(10,100,10).

Terafuga-hundred

100^^^^100

Gigangol

E100#100#100#100

The next major extension to the googol of Saibian’s. This number’s name is short for gigantic googol, because it’s, well, gigantic. Here’s how to visualize gigangol:

Stage 1 = 10^10^10.........^10^10^100 with 100 10s = grangol

Stage 2 = 10^10^10.........^10^10^100 with stage 1 10s = grangoldex

Stage 3 = 10^10^10.........^10^10^100 with stage 2 10s = grangoldudex

....

Gigangol is stage stage stage ... ... ... stage 100 with 100 stages. This number is comparable to a geegol, just as grangol is to giggol and greagol is to gaggol.

Pentofaxul

200!3

This is the pentational factorial of 200, equal to 200^^^199^^^198^^^197.......^^^3^^^2^^^1.

Gigangolgong

E100,000#100,000#100,000#100,000

Great Graham

3^^^^^3

This number is equal to three heptated to the third. It is a number coined on Googology Wiki whose page was deleted for having no sources. It was created on the analogy gross:great gross::Graham's number:? The author probably mistook Graham's number to be 3^^^^3.

Geegolplex

10^^^^10^^^^100

In the stages used to calculate a gaggol and geegol, this is stage stage stage......stage 1 with a geegol stages.

Gigangoltetrex

E100#100#100#100#2

With Saibian’s numbers, we constantly need to define new suffixes. This is stage stage stage.....stage 100 with a gigangol stages.

Tetra-exaxis

10^^^^^4

After that we have eptaxis, octaxis, ennaxis, and dekaxis, followed by endekaxis and dodekaxis.

Gigangoldutetrex

E100#100#100#100#3

Tripent

5^^^^^5 or {5,5,5}

A tripent is 5 heptated to the fifth, or a linear array of three fives. Jonathan Bowers also defines a trisept and a tridecal (both of which we will see in a bit). Aarex completes this tri-x series with trihex, trioct, and triennet (also known as trienn).

For a visual representation of tripent which I made, click here.

Deka-exaksys/petafuga-ten

10^^^^^10

This is ten heptated to the tenth.

Gigol

10^^^^^100 = 10{5}100

A gigol is equal to ten heptated to the hundredth, expressible as the unwieldy 10^^^^^100 using up-arrow notation or as 10{5}100 using Bowers' operator notation. Here’s how to represent a gigol:

Use the stages for the gaggol and geegol.

Gigol =

Stage Stage Stage Stage Stage.....................Stage 1

where there are Stage Stage Stage Stage Stage.....................Stage 1 stages

where there are Stage Stage Stage Stage Stage.....................Stage 1 stages

where there are Stage Stage Stage Stage Stage.....................Stage stages

: : : : :

: : : : :

: : : : :

: : : : :

: : : : :

: : : : :

: : : : :

where there are Stage 1 stages

with 100 lines. YIKES! This representation is truly insane.

Gorgegol

E100#100#100#100#100

This enormous number is short for gorged googol, which means “bursting with fullness”. You can imagine it like so:

Use the stages for the greagol and gigangol.

Gorgegol =

Stage Stage Stage Stage Stage.....................Stage 100

where there are Stage Stage Stage Stage Stage.....................Stage 100 stages

where there are Stage Stage Stage Stage Stage.....................Stage 100 stages

where there are Stage Stage Stage Stage Stage.....................Stage 100 stages

: : : : :

: : : : :

: : : : :

: : : : :

: : : : :

: : : : :

: : : : :

where there are Stage Stage Stage (100 stages).....Stage 100 (a.k.a. gigangol) stages

with 100 lines. This representation really makes this number feel gorged!

Hexofaxul

200!4

The hexational factorial of 200, equal to 200^^^^199^^^^198 ... ... ^^^^2.

Gigolplex

10{5}10{5}100

This number is like the representation of a gigol, but with a gigol lines instead of merely 100!!

Gorgegolpentex

E100#100#100#100#100#2

This is like a gorgegol, but its representation has a gorgegol lines. A gorgegoldupentex’s representation has a gorgegolpentex lines.

Goggol

10{6}100 or {10,100,6}

A goggol is equal to ten octated to the hundredth, yet another Boweian googolism. I'm going to leave it to you to imagine this insane number - keep in mind that it's still rather small amongst the googolisms though.

Gulgol

E100#100#100#100#100#100

This number is comparable to a goggol, and it’s short for *gulp* googol. Once again, I’ll leave it to you to imagine this horrendously huge number. Hint: It’s also a gorgegol-99-pentex.

Gulgolhex

E100#100#100#100#100#100#2

We take advantage of the prefix hexa- and use it to create -hex, simpler than the redundant-sounding -hexex.

Trisept

7{7}7 = {7,7,7}

Three sevens in a linear array - this mind-boggling number is equal to 7 enneated to the 7th.

Gagol

10{7}100 or {10,100,7}

A gagol is ten enneated to the hundredth, Bowers' last number of the form g-(vowel)-(g or gg)ol.

Gaspgol

E100#100#100#100#100#100#100

This epic number is short for *gasp* googol. It is comparable to the gagol. Even visual representations now are hard to get your head around, and cumbersome to make.

Lower bound for q(5)

A(9,A(8,A(8,255)))

q(x) is a fast growing function which involves Laver tables - it's defined as the smallest number n such that p(n) (which is the period of a the first row of a size n Laver table) = 2^x, and q(x) is the smallest y such that p(y) = 2^x. The first four values of q(x) are 2, 3, 5, and 6, but the existence of q(5) hasn't been confirmed - if it exists, it's at least this big. In up-arrow notation, this lower bound is approximately 2{7}2{6}2{6}255, placing it between a gagol and a gagolplex.

Gagolplex

10{7}10{7}100

Gaspgolheptex

E100#100#100#100#100#100#100#2

Boogaten

10{8}10

Also the old value of Bowers' tridecal, a few entries later.

Ginorgol

E100#100#100#100#100#100#100#100

As insane as it looks. This number is short for ginormous googol and is comparable to 10{8}100, although Bowers doesn't have a term for that value.

Ginorgoltetrex

E100#100#100#(E100#100#100#100#100#100#100#100)

This is an unusual number coined by Sbiis Saibian using ginorgol + the -tetrex suffix. Because the -tetrex suffix has basically no effect on numbers this big, it happens to be a salad number. The only reason this number is on this list is because it's Sbiis Saibian's googolism number 1000 on his Hyper-E article - on all his Extensible-E number lists he numbers his googolisms.

Ginorgoloctex

E100#100#100#100#100#100#100#100#2

Gaxoogol

g(10,100,10) = 10{9}100

Could be a continuation of giggol, gaggol, and geegol, but nah. Just a Joycian googologism that is part of the gamoogol (the next number in the group) family. The number is comparable to Saibian's new number, gargantuul.

Gargantuul

E100#100#100#100#100#100#100#100#100

A gargantuul, short for gargantuan googol, is one of Sbiis Saibian more recent googolisms. It's approximately equal to 10{9}100, and it can be compactly written as E100##9 using the ## operator.

Gargantuulennex

E100#100#100#100#100#100#100#100#100#2

The Multi-Arrow Range

10{10}10 ~ 10{10^100}102

(10{10}10 ~ order-type w)

Entries: 21

10{10}10 or {10,10,10}

This googolism (pronounced "TRY-da-cal") was coined by Jonathan Bowers. It's equal to three tens in a linear array, or ten dodecated to the hundredth. It's notable because Bowers considers anything bigger than this an Infinity Scraper, so you could say the real big numbers start here. This number is very horrendously huge - it expands to

10{9}10{9}10{9}10{9}10{9}10{9}10{9}10{9}10{9}10

= 10{9}10{9}10{9}10{9}10{9}10{9}10{9}10{9}10{8}10{8}10{8}10{8}10{8}10{8}10{8}10{8}10{8}10

= 10{9}10{9}10{9}10{9}10{9}10{9}10{9}10{9}10{8}10{8}10{8}10{8}10{8}10{8}10{8}10{8}10{7}10{7}10{7}10

{7}10{7}10{7}10{7}10{7}10{7}10

etc.

The expression grows very rapidly and requires insane levels of iteration.

This is also the second and last Bowersism Joyce gets right, the other being tritri.

With Saibian's -logue, -taxis, -petaxis, -exaxis, etc. series, this number can be called deka-endekaxis.

Joycian boogol

g(11,100,10) = 10{10}100

Joyce attempted to define the boogol, and once again failed. Luckily, this time he only made a small mistake. He mixed up the order; it should be g(101,10,100).

E100#100#100#100#100#100#100#100#100#100

The googondol is the largest of Sbiis Saibian's primitive googol extensions. It's short for "googondo googol" - googondo is a slang word for gigantic (Urban Dictionary). This number is comparable to Bowers' tridecal and is approximately equal to 10{10}101. It can be compactly written as E100##10 (see gugold for more on that)

Googondoldecex

E100#100#100#100#100#100#100#100#100#100#2

g(1)

2{12}3

Little Graham was the original version of Graham's number. It is calculated as g(x) = 2{g(x-1)}3, and g(1) = 2{12}3. The value simplifies to 2{11}4 = 2{10}2{10}4.

3{27}3

This is a number that shows up mostly when working with chained arrows of 3, like in conway's tetratri (seen later). It can be expressed as 3->3->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![1] = 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![1] using the very beginning of Hollom's hyperfactorial arrays since x![1] = 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