Well done everybody, at the start of the thread I thought I had a problem to solve, but it was already solved by the time I woke up. There are indeed an infinite number of games, as proved by the simple case with 2 pieces each left.

As for the infinitely long game, I agree with Wil. it is possible in theory to have an infintely long game. The idea that it's probability approaches zero only shows us that we have no possibility of ever seeing the game to completion. This, of course, is the point.