Whatever largest number you can express in your system, I can represent a larger one in only one bit, using the following specification.

0=your largest number 1=your largest number + 1

To be pedantic, that is a instance of the Berry paradox [1] and no you can not [2] as that would be a violation of Godel's incompleteness theorems.

edit: To clarify further, you could create a new formal language L+ that axiomatically defines 0 as "largest number according to L", but that would no longer be L, it would be L+. For any given language with rules at this level of power you could not make that statement without creating a new language with even more powerful rules i.e. each specific set of rules is capped, you need to add more rules to increase that cap, but that is a different language.

[1] https://en.wikipedia.org/wiki/Berry_paradox

[2] https://terrytao.wordpress.com/2010/11/02/the-no-self-defeat...