The total number of legal games has also been explored by collaboration between Matthieu Walraet and John Tromp:
They constructively establish that there are at least 10^10^100 (a googolplex) legal games of go.
The googolplex is an absurdly large number, since it has a googol (10^100) digits. I believe the estimate for the number of atoms in the universe is around 10^80. The age of the universe in seconds is less than 10^18. Thus, it seems that there is not enough space nor time to even write down the exact number of go games, much less to precisely compute it.
Earlier draft: