What are some good methods to estimate the maximum possible Grundy Number that can occur while evaluating an impartial game, (assuming that the game has no subgames)?
(This helps in estimating the size of data structures to use when calculating MEX).
I am not sure precisely what you are asking, but the nimbers or Grundy numbers have the addition function $$\alpha \oplus \beta = \mathsf{mex}(\{\alpha^\prime \oplus \beta : \alpha^\prime \lt \alpha \} \cup \{\alpha \oplus \beta^\prime : \beta^\prime \lt \beta \}$$ which is closed on the set $\{0,1,2,\ldots,2^{k}-1\}$ for any integer $k$.
So you may want to use a bound of the next power of $2$ of the all the components of your impartial game.
In the unlikely event you also want to multiply nimbers, that is closed on the set $\{0,1,2,\ldots,2^{2^n}-1\}$ for any integer $n$, so you have a bound which is the next number which is $2$ to the power of a power of $2$.