There are estimated to be about possible positions on a board (Flammenkamp). The number of -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.