There are only 64 spaces and there are 16 pieces each. How many possible move combinations are there?

Chess is infinite: There are 400 different positions after each player makes one move apiece. There are 72,084 positions after two moves apiece. There are 9+ million positions after three moves apiece. There are 288+ billion different possible positions after four moves apiece. There are more 40-move games on Level-1 than the number of electrons in our universe. There are more game-trees of Chess than the number of galaxies (100+ billion), and more openings, defences, gambits, etc. than the number of quarks in our universe! --Chesmayne

Even when we start with something so simple, we can see how quickly complexity explodes.

Even when we start with something so simple, we can see how quickly complexity explodes.