"Conceding" means passing. If you don't pass, you have to play. If you play when the game is effectively over, you either play in your opponent's territory and get captured, or your own, which reduces your score. If you play in your own territory enough then you can actually end up losing your eyes, and then be captured.
I guess the remark comes from the "perhaps infinite" proof game. I don't know why they put "perhaps infinite" there as most Go rulesets have rules against repeating board positions.
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.
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)
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