http://en.wikipedia.org/wiki/Shannon_number  refers to math research on the topic.  10^43 is offered as the number of posisitions that pieces can be put into.  (10 followed by 43 0's - a little larger than the national debt).

BUT that includes both legal and illegal posisions, which overestimate, but, that's only posisitions.  There can be many ways to arrive at the same posistion which would make different games - so in that sense it undercounts the number of possible games.

I submit that the reason it is not infinite is due to the '50 move rule' which draws a game after no action for 50 moves.  otherwise you could have some trival back and forth moving that made an infinite game.....

maybe my sophmore student's math teacher would know.

Obviously it is not infinite,because in order 2 be that ,a person would need a board with  more than 64 squares.So with the mathes.The variation would need 2 be greater and the squares more, though by how much is debateable,as it would have 2 add up ultimatly 2 an infinite number.I am not sure if that would be concievable or not.