Can a finite game be infinitely complex

This is a problem that's been bugging me for a bit. If you give an answer, please back it up.

Take a game with N finite steps/turns. Is it possible for such a game to be infinitely complex-rather, have an infinite amount of possible moves?

For A fixed number of moves, the complexity is finite. For a game like, say, chess, there are conceivably games with an infinite number of turns, so the complexity (number of possible different combinations of moves?) is also infinite.

Well infinite isn't really a number. It can be hugely complex, but when it come down to it, there is a set number that can be done, the number is just huge. For example, Chess. There are TOOOOONS of possible playouts, but they could be calculated. Nothing can really be infinitely complex unless it is always changing and expanding, like the Universe.