A mathematically established lower bound on the longest possible go game is 10^10^48 [1]. While this isn't infinite, it's certainly large enough to be considered infinite in practice.
"Graham and Rothschild (1971) also provided a lower limit by showing that N must be at least 6. More recently, Exoo (2003) has shown that N* must be at least 11 and provides experimental evidence suggesting that it is actually even larger."*
In this case, I think it is safe to claim that the answer is at least 361!/8, though (but that may already include many truly silly games with suicidal moves in the opening or games that continue way past the time experienced players think they are over)