[Someone]

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)