If you play chess, you might think that there is an infinite number of chess games, considering the players can move their pieces around forever without taking any opposing pieces. But this is more about the possible moves that can be made.
At the start of chess, the starting player has one of 20 possible moves to choose from with his pawns. The opposing player then has 20 moves with which he can respond. That means after the first turn there can be 400 different positions.
One more turn and the number of positions reaches about 20,000. One Claude Shannon worked out the total number of possibilities, which is the astronomical number above, now called the Shannon number (even higher than googol). If you’re a chess player, this still might seem a little odd, but that’s because the vast majority of these games aren’t good games, and generally speaking many of the bad games will never be played.