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RichardKennaway comments on Probabilities Small Enough To Ignore: An attack on Pascal's Mugging - Less Wrong Discussion
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FInite approximations to the St. Petersburg lottery have unbounded values. The sequence does not converge to a limit.
In contrast, a sequence of individual gambles with expectations 1, 1/2, 1/4, etc. does have a limit, and it is reasonable to allow the idealised infinite sequence of them a place in the set of lotteries.
You might as well ask why the sum of an infinite number of ones doesn't exist. There are ways of extending the real numbers with various sorts of infinite numbers, but they are extensions. The real numbers do not include them. The difficulty of devising an extension that allows for the convergence of all infinite sums is not an argument that the real numbers should be bounded.
They have unbounded expected values, that doesn't mean the St. Petersburg lottery can't exist, only that its expected value doesn't.