Proof that grahams number is the largest number?

Answer 1

No such proof can exist within consistent axioms - to add 1 to Graham's number would generate a larger number, and so, by this counterexample, Graham's number cannot be the largest.

It is the largest used in a mathematical proof, because all proofs have been noted in a proof 'by exhaustion', in which all cases (in this case, proofs) have been verified to have smaller constants .

Answer 2

It is so big that it literally cannot be written down, if each digit of graham's number were written down so small that each digit took up 1 plank volume, the written number would take up a space larger than the observable universe.