Trialogue
Trialogue (also School House Rock Number)
10^10^10
A trialogue is a googolism coined by Sbiis Saibian equal to one followed by ten billion zeroes. It can be written as E1#3 or E10#2 in Hyper-E, and 10^^3 in up-arrow notation, and it is the third member of the -logue series. This number is pretty big, but still small enough to represent numbers of possibilities or odds of a rare event. It can still be realistically written out, but it would take up lots of space, physically or digitally (especially physically, so writing this digitally is a far better option). Also, what would be the point of writing out this number if it’s just 1 and ten billion zeroes? Nevertheless, it's still feasibly doable.
The whole sequence of x^x^x from 1 up to a trialogue is an interesting sequence:
1^^3 = 1 is a trivial number.
2^^3 = 16 cannot be recognized at once but can be visualized
3^^3 = 7,625,597,484,987 can't be visualized but can be understood with real life examples
4^^3 is difficult to understand with real life examples but can be written to take up a line of space
5^^3 takes up a paragraph of space
6^^3 takes up a few pages of space
7^^3 takes up about a few chapters worth of space
8^^3 takes up about a book's worth of space
9^^3 takes up about 33 books worth of space
10^^3 takes up about a thousand books worth of space
As you can see, each of those numbers is a different level of space it takes to write them, ranging from one glyph to a thousand books. Now think about that for a minute. Here "a book" doesn't mean an average sized book. It means the size of a dictionary, with about a thousand pages and ten thousand digits per page. Imagine a big room filled with bookshelves to hold dictionaries, enough space to hold a thousand dictionary-sized books. Now imagine the each book having zeros and zeros and zeros and zeros, lots and lots of zeros, ten million per book. The only nonzero digit in any of those books is the one on the left end of the first line of the first page of the first book. It's something that definitely can be done, but is really inconvenient.
I would, however, be interested by the idea of a library building, perhaps in a university, to house, say, thirty thousand books. Among those books and books is a floor of the building, the topmost among, I don't know, seventeen floors, because seventeen is an awesome number. This floor seventeen would be called "The Power Floor", and would house exactly one thousand thirty four books. Each book would have a black cover with, in golden Times New Roman font, the title 1^1^1 THROUGH 8^8^8 for the first book, 9^9^9 VOLUME 1 through 9^9^9 VOLUME 33 for the next 33 books, and 10^10^10 VOLUME 1 through 10^10^10 VOLUME 1000 for the other 1000.
The first book would be a bit thicker than the others, and would start with a single page that starts with a single digit 1, and then takes a line break and says 16. Then the next line would be devoted to the digits 7,625,597,484,987, with or without commas, I don't know. After that, two lines of space would hold the one hundred fifty four digits that make up 4^4^4. Then, a paragraph worth of text would hold the digits of 5^5^5. That would be the end of the first page. Then, pages two, three, four and five, would hold the tens of thousands of digits of 6^6^6. However, 7^7^7 would have to take up seventy pages. The last page of 7^7^7 would be gray, to make it easy for readers to find where the digits of 8^8^8 start. Then, the whole rest of the book, one thousand five hundred sixteen pages, is devoted to 8^8^8.
The rest of the books are a lot more mundane. The second through thirty-fourth books contain the digits of 9^9^9, with eleven to twelve million digits per book. They would lurk on two shelves, on a manufactured plastic bookshelf (ok, maybe wooden). Then, the 35th through 1034th will be exactly one thousand books, stored on perhaps ten bookshelves, five shelves per bookshelf, twenty books per shelf. The first of those books, the 35th total, has a page with a one and then ten thousand zeros, and all other 999 pages are ten thousand zeros each. Then, the other 999 books, number 36 through 1035, would be nine hundred ninety nine identical books, with one thousand identical pages of ten thousand zeros per page. All those to store the digits of ten raised to the ten-billionth power. Not many people would look at them, and most of the books would likely remain untouched.
So why bother? I would say that's mostly to prove that a feat like this is easily possible. Perhaps the ten billion digits are stored somewhere on the Internet, but using the Internet to store is "cheating" since it doesn't reflect the space requirement. Not cheating because I hate the Internet (which I don't), but cheating because it doesn't reflect the ideas of physical space needed to store all these digits on real, tangible sheets of paper. not just as bits on a hard drive that might take up a cubic inch of space at the VERY most. Those bits would take a microscope to see and cannot be interpreted by someone who is not a computer specialist. Those bits are only projected as digits as pixels on a computer screen. Paper, on the other hand, has all the digits directly visible on a sheet of paper, easily tangible and visible directly - you can directly see the digfits on the place where the digits are stored instead of merely seeing bits of data projected on a computer screen, and probably only part of them even projected. This goes to show that I just think it's a good idea to store digits of large numbers in books, to bring to people the sense of vastness that is more easily reflected with real books than with a computer screen.
Aside from that, a trialogue is indirectly mentioned in the School House Rock song "My Hero, Zero", which discusses the importance of the number 0:
Place a zero after one,
and you've got yourself a ten --
see how important that is!
When you run out of digits
you can start all over again --
see how convenient that is!
That's why with only ten digits, including zero,
you can count as high you could ever go --
forever, towards infinity.
No-one ever gets there, but you could try ...
with ten billion zeros.
It doesn't say what exactly you would do with those ten billion zeroes, but it's pretty clear that those zeroes would be used to make a very large number; with all those zeroes, you could use them to get 1 followed by 10 billion zeroes, or even a bigger number. This is supported when as the last few lines listed above play, the screen shows a pyramid starting with 9, 80, 700, 6000, 50000, and as it goes down the pyramid the screen becomes filled with tiny zeros, clearly showing that the ten billion zeros would be used to write a certain very large number. The song in general focuses a lot on very large numbers. Robert Munafo claims that to his knowledge, it's the largest number indirectly mentioned in a published piece of music.