User blog comment:Deedlit11/Googology in Magic: the Gathering/@comment-5150073-20130412144609/@comment-5150073-20130413110858

Well, any game can be programmed at the Turing machine (through in the abstact, primitive form) with the finite number of states. Why in this game possible to generate function with arbitrary level of recursion, if this game itself represented by some recursive ordinal? Recursive ordinal can't have other recursive ordinals above it in those sets.

For example, we can program some game with 500 states on the Turing machine. The most complicated game (with the largest time complexity) that we can program with 500 states is represented by the ordinal w_1^CK[500], 500-th member of the fundamental sequence to the Church-Kleene ordinal. We know that every member of that sequence is a recursive ordinal, so no game can be represented by w_1^CK itself.