The "name the biggest number game" asks two players to write down a number secretly, and the winner is the person who wrote down the larger number. The game commonly allows players to write down functions evaluated at a point, so $2^{2^{2^{2}}}$ would also be an acceptable thing to write down.
The value of the Busy Beaver function, $BB(x)$, cannot be determined (in ZFC, or any reasonable consistent axiomatic system) for large values of $x$. In particular, $BB(10^4)$ cannot be determined as per this paper. However, this doesn't mean that we cannot compare values of the Busy Beaver function. For example, we can prove that $BB(x)$ is strictly monotonic.
Lets suppose that we allow players to write down expressions involving compositions of elementary functions, natural numbers, and the Busy Beaver function. Are there two expressions that the two players can write down such we can prove in ZFC that determining the winner in ZFC is impossible (assuming ZFC is consistent)?
EDIT: Originally this question said “... arbitrary combinations of computable functions, natural numbers, and the Busy Beaver function.”
If we let $f(x)$ take on the value of $3$ if $BB(x) >$ [something ungodly large and inexpressible on this website] and $7$ if it’s not, then $f(10^4)$ and $6$ are incomparable.
This doesn’t satisfy me, largely because $f$ isn’t a reasonable function for someone to use in this game. I don’t see how to phrase my intuition about this though, so I’ve restricted the question to avoid piecewise functions.