For what may or may not be an example, Graham's number is astonishingly huge, but math world (http://mathworld.wolfram.com/GrahamsNumber.html) claims the answer to problem it is an upper bound for may even be larger than 11:
"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)
"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)