Are there any combinatorial games with odd order (under the usual addition of combinatorial games), apart from $0$?
In Are there combinatorial games of finite order different from $1$ or $2$? I asked about games of finite order greater than $2$, and was given a really nice example of a game of order $4$ (and I feel that the given example can probably be generalized to get games whose orders are higher powers of $2$), but in the comments to that answer it was indicated that there might not be any of odd order.
No, there are no (non-zero) games of odd order, but you can indeed construct games whose order are arbitrary powers of two. There is a beautiful proof of these facts in "Combinatorial Game Theory" by A. Siegel, chapter III section 3.