The expected win is infinite because very rare scenarios have huge payouts. However, any one play of the game has a 50% chance of paying out nothing. If I get to play the game a large enough number of times for the rare scenarios to actually occur, then I would be willing to pay a higher price than if I only got one shot at it.
However! Suppose you were allowed to modify things ever so slightly. Suppose you get to say, "Sure, I'll pay $1 million to play. But instead of playing once and taking my winnings, I want to play a million times, and only take one millionth of whatever I win each time." You haven't done anything to change the average payout of the game. But you've made it a much better idea to play, by reducing the variance.
That's what it means to say the game has infinite expected value. You really should be willing to pay anything to play it, as long as you're allowed to repeat the game over and over.
Of course, if you're stuck with playing the game just once, then lots of games look like a bad idea despite having a positive expected value. The St. Petersburg game is hardly alone here. For example, suppose you have a 1-in-a-million chance of winning $10 billion, but it costs $1000 to play. Should you play? The expected value is good, but I don't think most people would touch it. Not much of a paradox there.
I'd happily take those odds.
Not quite. The expected win is infinite because of the one single scenario where the games continues on an infinite basis.
The average win is approx $46 if the bankroll is all the money in the world.
Arguably you could model this with hidden variables. So: A mathematician might go to vegas and spend 1,000$ gamboling with the expectation that they will lose that money. With novelty being the reason that becomes a reasonable trade-off.
In that context maximizing expected utility becomes hard because you can't accurately model your utility function.