Rayo's Number

This number is the result of a "large number duel"[1] between MIT Associate Professor Agustin Rayo (The Mexican Multiplier) and Princeton Associate Professor Adam N. Elga (Dr. Evil)[2]. Possible the world record largest number by some standards.

[1] https://www.scottaaronson.com/writings/bignumbers.html

[2] http://tech.mit.edu/V126/N64/64largenumber.html

So its clear:

It's the smallest positive integer bigger than any finite positive integer, "named" (defined, so some formula without explicitly writing it out) by an expression in the language of first order set theory with a googol symbols (10^100) or less.

I remember reading busy beaver numbers can be defined in a language with millions of symbols, so this number is pretty impressive.