Largest number of 1's that can be printed by a machine with N states that halts.
https://en.wikipedia.org/wiki/Busy_beaver
http://www.scottaaronson.com/writings/bignumbers.html
Function grows really fast, and only finitely many numbers of it can be proved.
Only finitely many 3-4 are known, 7,918 is independent from ZFC (assuming some likely hypothesis): http://www.scottaaronson.com/writings/bignumbers.html http://www.scottaaronson.com/blog/?p=2725