One thing we do know about the Grundy number of a position is that it is less than or equal to the ordinal representing the maximum length of the game starting from that position; to be precise, define $\ell(p)$ to be 0 if the position $p$ is an ending position, and

$$
\ell(p) = \sup_{q \in p}\{\ell(q)+1 \}
$$

otherwise. (by $q \in p$ I mean that from position $p$ the acting player can move to position $q$)  Then the Grundy number $G(p) \le \ell(p)$.

For Sylver Coinage, $\ell$ of the starting position is $\omega^2$, and $\ell$ of any other position is strictly less than $\omega^2$.  So the Grundy number at most $\omega^2$ for the starting position, and strictly less for any other position.