What lies beyond large numbers? Tetration, pentation, non-integer hyperoperators, Knuth up-arrow notation, Conway chained-arrow notation, Ackermann function, Bird's array notation, ordinals, cardinals, hyperreals, surreals...

[Patashu]

If addition is hyperoperator 1, multiplication is hyperoperator 2 and exponentiation is hyperoperator 3, tetration is hyperoperator 4 and pentation is hyperoperator 5. These operators are notable for being easy to define for integers, but extremely hard to define for real and complex numbers. (Speaking of which, complex numbers when? Vectors/matricies when? Etc...)

• https://en.wikipedia.org/wiki/Tetration
• http://www.tetration.org/
• https://en.wikipedia.org/wiki/Pentation

Like exponentiation, tetration has two inverses - the super-root and the super-logarithm:

• https://en.wikipedia.org/wiki/Tetration#Inverse_operations

With this hierarchy in mind, we could ask what non-natural number hyperoperators look like - what's hyperoperator 0, hyperoperator -1, hyperoperator 0.5, hyperoperator 1.5, etc?

• https://en.wikipedia.org/wiki/Hyperoperation
• http://googology.wikia.com/wiki/Hyper_operator
• http://andydude.github.io/tetration/archives/tetration2/hyper.html
• https://sites.google.com/site/pointlesslargenumberstuff/home/2/weakoperators

To encode very large numbers, some formats and functions exist:

• https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation
• https://en.wikipedia.org/wiki/Conway_chained_arrow_notation
• https://en.wikipedia.org/wiki/Ackermann_function

The inverse ackermann function is notable for being one of the slowest growing functions in existence:

• https://en.wikipedia.org/wiki/Ackermann_function#Inverse

And the biggest and most absurdly growing notation of all is Bird's array notation:

• http://googology.wikia.com/wiki/Introduction_to_Bird%27s_array_notation
• http://www.mrob.com/users/chrisb/

And can we go even further?

• http://forums.xkcd.com/viewtopic.php?f=14&t=7469&start=800 'My number is bigger!', especially http://forums.xkcd.com/viewtopic.php?p=2218880&sid=7158c87de2c2c31ee8c7c934d46cdb63#p2218880 and following page, http://forums.xkcd.com/viewtopic.php?p=2596056#p2596056 , http://forums.xkcd.com/viewtopic.php?p=3254229#p3254229 (EliezerYudkowsky's massive computable number) and following page
• http://forums.xkcd.com/viewtopic.php?f=14&t=7469&p=4280817#p4280817 Finally I added my own post for funsies
• http://googology.wikia.com/wiki/Largest_valid_googologism Rayo's Number, BIG FOOT and beyond
• http://googology.wikia.com/wiki/Greedy_clique_sequence and other large combinatronical numbers
• https://itaibn.wordpress.com/2017/11/09/prefix-free-codes-and-ordinals/ a way to represent all numbers?
• https://en.wikipedia.org/wiki/Large_numbers 'Large numbers'
• http://djm.cc/dmoews.html http://djm.cc/bignum-results.txt http://djm.cc/marxen-comments.txt http://djm.cc/ralph-loader.tar December 2001 BIGNUM BAKEOFF contest
• http://googology.wikia.com/wiki/Fast-growing_hierarchy http://googology.wikia.com/wiki/Introduction_to_the_fast-growing_hierarchy
• http://zoologia.biologia.uasnet.mx/ydras/hydra%209.pdf http://fleischer.selfip.com/Paper/hydra_fun07.pdf https://link.springer.com/chapter/10.1007/978-3-540-72914-3_14 Kirby-Paris Hydra and Buchholz hydra and Die Another Day

Finally we reach ordinal and cardinal numbers, transfinite numbers representing sizes and kinds of infinity that can grow to absurd amounts. We've now stepped beyond the realm of numbers that even remotely make sense. Hyperreal/surreal numbers are another alternative - they make infinite and infinitesimal numbers quantities that can be further added/multiplied like any other number.

• https://steamcommunity.com/app/432380/discussions/0/135508292190387402/?tscn=1486172196 (many tutorials and articles curated here)

[Patashu]

https://github.com/Patashu/break_eternity.js might be more appropriate for this library :p

[Patashu]

Patashu/break_eternity.js#46