It is quite large
How many unique games of chess can be played??-The Shannon number
Fascinating topic. I have a couple blog posts on it, too: http://wp.me/p6NNod-5D and http://wp.me/p6NNod-5V

I feel the far more interesting question is how many legal positions there are (because this is a much smaller number, and knowledge of which positions lead to which positions by legal moves describes chess in a much more economical way than thinking of all possible games.
For example a 32-piece tablebase of chess would be all a perfect player required and is a database of positions and their relationships (recursively used to calculate the value of each position and the number of moves to a forced mate if there is one).
A smaller representation of perfect chess would be a strategy for each player to achieve the best result from any position and a tablebase of only those positions that are reachable from those strategies against any opponent. Most positions can be ignored as irrelevant (requiring both sides to blunder to be reached).
Optimistically, the latter might be as few as 10^30 positions (by hand-waving analogy with checkers and accurate estimates of the number of legal positions as about 10^47). It is conceivable that digital computers will eventually be able to cope with this large a state space, while the full 10^47 would require multiple planets!
How many unique games of chess can be played?? A quadrillion, maybe a googol (10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000) well it's more than that. A close estimation of the number was found in 1950 by the, "father of information theory", Claude Shannon. He estimated the lower bound on the game-tree complexity of chess to be about 10120. In his paper "Programming a Computer for Playing Chess " he explains:
With chess it is possible, in principle, to play a perfect game or construct a machine to do so as follows: One considers in a given position all possible moves, then all moves for the opponent, etc., to the end of the game (in each variation). The end must occur, by the rules of the games after a finite number of moves (remembering the 50 move drawing rule).Each of these variations ends in win, loss or draw. By working backward from the end one can determine whether there is a forced win, the position is a draw or is lost. It is easy to show, however, even with the high computing speed available in electronic calculators this computation is impractical. In typical chess positions there will be of the order of 30 legal moves. The number holds fairly constant until the game is nearly finished as shown [...] by De Groot, who averaged the number of legal moves in a large number of master games. Thus a move for White and then one for Black gives about 103 possibilities. A typical game lasts about 40 moves to resignation of one party. This is conservative for our calculation since the machine would calculate out to checkmate, not resignation. However, even at this figure there will be 10120 variations to be calculated from the initial position. A machine operating at the rate of one variation per micro-second would require over 1090 years to calculate the first move!
Later Dutch computer scientist Victor Allis estimated it to be closer to 10123
This is a huge number. For comparison note that some have estimated that the total number of atoms in the observable universe is somewhere around 1079 to 1081