user5635: 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)


user5809: According to my calculations is about 10 ^ 134 different variants of the game
http://jknow.republika.pl/chessexplorer/szachy.html

user2792: One simple argument that the number of chess games is finite could be as follows. 
Due to the 50-move rule, any 50-move subsequence of a given chess game will contain at least one capture or a pawn move. Since there are finitely many pieces on the board, and since pawns can move only finitely many times during a game, the number of moves in a chess game has a finite bound. Since in each move, there are only finitely many possibilities, the numbers of all games is finite.
Note that this argument is almost useless if one wants to get an estimate on the number of possible games. If for nothing else, the only thing I use above is the 50-move rule and how the pieces move, so the repetitions are allowed (max. 50-fold repetitions, of course). Hence, the argument is just theoretical, not practical.

user5852: At some point you'll run out of combinations. So the answer is basically no.

user825: Q1: Yes. The total number of chess games can be considered infinite for all practical purposes. We don't have the technology to brute force over the first 13 moves from the initial position.
Q2: The actual numbers all the way up to depth 13 is known. The exact number of possible positions for the 10th moves is 69,352,859,712,417. Read this Wikipedia article for more details.
There is an attempt for depth 14 but so far the calculation after months and months is still running.

user3237: 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.

5635: <p>This question is somewhat related to <a href="https://chess.stackexchange.com/questions/5135/can-the-total-number-of-possible-wins-draws-losses-be-calculated">Can the total number of possible wins/draws/losses be calculated?</a>, but slightly different. </p>

<p>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". </p>

<p>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?)</p>

<hr>

<p>Curiously, Wikipedia <a href="https://en.wikipedia.org/wiki/Go_game" rel="noreferrer">seems to be implying</a> that the number of games can be estimated: </p>

<blockquote>
  <p>the number of possible games [in Go] is vast (10<sup>761</sup> compared, for example, to the 10<sup>120</sup> possible in chess)</p>
</blockquote>
 5809: <p>According to my calculations is about 10 ^ 134 different variants of the game
<a href="http://jknow.republika.pl/chessexplorer/szachy.html" rel="nofollow">http://jknow.republika.pl/chessexplorer/szachy.html</a></p>
 2792: <p>One simple argument that the number of chess games is finite could be as follows. </p>

<p>Due to the 50-move rule, any 50-move subsequence of a given chess game will contain at least one capture or a pawn move. Since there are finitely many pieces on the board, and since pawns can move only finitely many times during a game, the number of moves in a chess game has a finite bound. Since in each move, there are only finitely many possibilities, the numbers of all games is finite.</p>

<p>Note that this argument is almost useless if one wants to get an estimate on the number of possible games. If for nothing else, the only thing I use above is the 50-move rule and how the pieces move, so the repetitions are allowed (max. 50-fold repetitions, of course). Hence, the argument is just theoretical, not practical.</p>
 5852: <p>At some point you'll run out of combinations. So the answer is basically no.</p>
 825: <p>Q1: Yes. The total number of chess games can be considered infinite for all practical purposes. We don't have the technology to brute force over the first 13 moves from the initial position.</p>

<p>Q2: The actual numbers all the way up to depth 13 is known. The exact number of possible positions for the 10th moves is 69,352,859,712,417. Read <a href="https://en.wikipedia.org/wiki/Shannon_number" rel="nofollow noreferrer">this Wikipedia article</a> for more details.</p>

<p>There is an attempt for depth 14 but so far the calculation after months and months is still running.</p>
 3237: <p>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.</p>

<p>So no, the number of possible chess games is not infinite.</p>

<p>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</p>

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

<p>Article 9.3 of FIDE Laws of Chess states that:</p>

<blockquote>
  <p>9.3   </p>
  
  <p>The game is drawn, upon a correct claim by a player having the move,
  if:</p>
  
  <ul>
  <li>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</li>
  <li>the last 50 moves by each playerhave been completed without the movement of any pawn and without any capture.</li>
  </ul>
</blockquote>

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

<p>But if you're not interested with the previous theoretical numbers:<br>
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<br>
As for the total number of legal positions, it's somewhere between 10^40 and 10^50.</p>
