Is there a short game $G \ne 0$ such that $G + G + G = 0$? It seems to me like such a game shouldn't exist, but I am unable to prove it. Can anyone give an example of such a game, or a proof that one doesn't exist?
Also, is there any short game $G$ such that an odd number of copies of $G$ summed together gives $0$? Note that if there is not, then any nilpotent game has order a power of 2.