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.

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, $\ell(p)$ is rather easy to determine. $\ell$ of the starting position is $\omega^2$; for any other position, the players will have selected some nonzero finite set of positive integers. Let $d$ be the GCD of all the numbers in the set, and let $m$ be the number of (not necessarily distinct) primes dividing $d$. Let $n$ be the number of possible next moves that are divisible by $d$. Then $\ell(p) = \omega m + n$. The Grundy number must be less than or equal to this ordinal.