[pzhon]

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I can set my total number of bets to an arbitrarily high number and my overall EV will increase exponentially with the number, but if I take the limit as the number goes to infinity, my EV goes to -100%.
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How do you reconcile this mathematically


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f(y) is not always the limit of f(x) as x goes to y. This fails when f is discontinuous at y. A lot of reasonably nice functions are discontinuous at infinity, like sin(x). It's something you have to look for whenever you use infinity in a mathematical model. Discontinuity at a value is a hint that the value is outside the applicable context of the model.

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what is the correct (ie highest EV) action with respect to the man's bets?

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First, E$ is not necessarily the same thing as E(Utility). You do not have to assume that the difference between being broke and having $1 billion is half of the difference between being broke and having $2 billion, so you do not have to find each toss better than not.

Second, there doesn't have to be an optimum. If the stranger says, "Tell me an integer, and I will give you that amount," there is no optimal strategy. If the stranger says, "Tell me a number strictly less than 1, and I will give you $1 with that probability," then there is no optimal strategy. There is neither a largest integer nor largest real number less than 1.