numebr of possible games

I have absolutly no idea how many games can be played but I can show that it is finite (at least within the rules)

the 3-fold repetition rule means that a position can be repeated 3 times (given that certain rights remain) but even then a position can only appear something like ten times? (twice if the right for four different castlings can happen black/white Queen side/King side) so that's eight an extra one for en passant (I think it would be safe to say that pawns cant't go in cycle to create the same position with this right reinstated: i.e a different pawn, given that they only move in one direction) and tenth time when it after the final castling and the enpassant right is lost and combined with the last two creates the same position with the rights 3 times.

if EVERY arrangement of pieces is reapeated 10 times in a row then you can only put them in a finite number of orders (the sequence is only finite)

of course:

- some positions are illegal (they can't be reached)

- only a few positions can appear the full ten times

- some games will end much sooner (due to one of the earier positions being a checkmate, stalemate etc)

- not every position can be turned in to every other position by one move

but this only reduces - an already finite - number of possible games proving that only finitly many games can be played

interesting question:

how many of these haven't been played yet?

Most of them. By a long shot.