Rayo's number

Rayo's number is an incredibly enormous (and possibly infinite) number that was created during a large number battle pitting Agustín Rayo against Adam Elga. Rayo's number is, in Rayo's own words, "the smallest number bigger than any finite number named by an expression in the language of first order set theory with a googol symbols or less."

Rayo and Elga's number duel was based on an article by Scott Aaronson.