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)