First off, you need to eliminate cycles; the number of games is infinite if cycles are allowed. Various superko rules address the issue of cycles. Both Japanese and Chinese rules do have some open issues regarding cycles. One solution is to simply forbid a move that recreates a previous board (positional superko). This rule makes the number of possible games finite.

This question has a conceptually easy answer: Simply complete Sloan's sequence A007565 (

http://www.research.att.com/~njas/sequences/A007565) to N = the longest game possible and sum the sequence. Since the longest possible game is somewhere between 10

^{48} and 2.08×10

^{170} moves, this might be a daunting task.

Sensei's Library is an excellent resource for matters concerning go. It's page on the number of possible go games,

http://senseis.xmp.net/?NumberOfPossibleGoGames, cites a paper by Tromp and Farnebäck,

* Combinatorics of Go* (

http://homepages.cwi.nl/~tromp/go/gostate.ps), which addresses the 17x17 board but could not quite address the 19x19 board.

The Sensei's Library page places an upper bound of 361^(2.08×10^170) on the number of possible games and a lower bound of 10^(10^48).