Go

There are estimated to be about $4.63\times 10^{170}$ possible positions on a $19\times19$ board (Flammenkamp). The number of $n$-move Go games are 1, 362, 130683, 47046242, ... (Sloane's A007565).

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 96, Feb. 1972.

Flammenkamp, A. ``A Short, Concise Ruleset of Go.'' http://www.uni-bielefeld.de/~achim/gorules.html.

Kraitchik, M. ``Go.'' §12.4 in Mathematical Recreations. New York: W. W. Norton, pp. 279-280, 1942.

Sloane, N. J. A. Sequence A007565/M5447 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.