Sylver's coinage is an example of an unbounded finite (if slightly modified) combinatorial impartial game. Quoth wikipedia:
The two players take turns naming positive integers that are not the sum of nonnegative multiples of previously named integers. After 1 is named, all positive integers can be expressed in this way: 1 = 1, 2 = 1 + 1, 3 = 1 + 1 + 1, etc., ending the game. The player who named 1 loses.*
This can be made to have the normal play convention if we make $1$ an illegal move.
In Conway's ONAG, it is shown that Grundy's number can be generalized to ordinal numbers for unbounded impartial games.
Is anything known about Grundy's ordinal for Sylver's Coinage and its various positions?
*Wikipedia contributors. "Sylver coinage." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 21 Oct. 2014. Web. 27 Mar. 2015.