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The natural conclusion is that the complexity of the problem depends on how much of the game you consider to be the `endgame'. Thus, the actual number of different chess games: 5

2) White mates 1) Black resigns 0) Stalemate -1) White resigns -2) Black mates

There are at least 3 others - there's - Black/White's clock runs out - game completion gets postponed for whatever reason and never resumed

i am not sure this is a very reasonable question without any restrictions. while this answer is humerous, i don't think anybody has addressed the fact that i can move a king back and forth between 3 squares infinitely many times.

You can't - after the board has been in the same position three times, the rules say it's a stalemate. This lets you calculate an upper bound which somebody did a few days ago.