Is the number of possible chess games infinite? This question is somewhat related to Can the total number of possible wins/draws/losses be calculated?, but slightly different. 

There is a recent TV show episode that claims that there are "more possible games of chess than atoms in the universe". They go on that "each possible move represents a different game, a different universe [..]"; "by the second move there are 72084 possible games, by the third -- 9 million, by the fourth --- 318 million". 

So is the total number of chess games infinite, for all practical purposes given human and technological limitations? And do the above numbers actually hold up to scrutiny? (i.e. What are the estimated possible games by, say, the 10th move?)



Curiously, Wikipedia seems to be implying that the number of games can be estimated: 


  the number of possible games [in Go] is vast (10761 compared, for example, to the 10120 possible in chess)
The maximum number of moves in a chess game is not infinite, it's 11797 plies = 5898 moves and a half. This is due to the fifty-move rule.

So no, the number of possible chess games is not infinite.

The maximum number of legal moves in a position is 218. So a crude upper bound for the number of possible chess games is 218^11797 = 10^27586

Wait, actually after fifty moves without any capture or pawn movement the players can also continue playing without claiming the draw...

Article 9.3 of FIDE Laws of Chess states that:


  9.3   
  
  The game is drawn, upon a correct claim by a player having the move,
  if:
  
  
  he writes his move, which cannot be changed, on his scoresheet and declares to the arbiter his intention to make this move which will
  result in the last 50 moves by each player having been made without
  the movement of any pawn and without any capture, or
  the last 50 moves by each playerhave been completed without the movement of any pawn and without any capture.
  


So I guess the number of possible chess games could be considered as infinite then...

But if you're not interested with the previous theoretical numbers:
The average number of legal moves in a position is around 35, and the average length of a chess game is around 40 moves = 80 plies, so an estimate of the number of "rational" chess games is 35^80 = 10^123
As for the total number of legal positions, it's somewhere between 10^40 and 10^50.