Suppose that at the beginning of the game, you decide to play no more than N turns. If you lose all your money by then, oh well; if you don't, you call it a day and go home.
After 1 turn, there's a 1/2 chance that you have 3 dollars; expected value = 3/2
2 turns, 1/4 chance that you have 9 dollars; expected value = (3/2)^2
3 turns, 1/8 chance of 27 dollars; E = (3/2)^3
4 turns, 1/16 chance of 81 dollars; E=(3/2)^4
...
N turns, 1/2^N chance of 3^N dollars; E=(3/2)^N
So the longer you decide to play, the higher your expected value is. But is a 1/2^100 chance of winning 3^100 dollars really better than a 1/2 chance of winning 3 dollars? Just because the expected value is higher, doesn't mean that you should keep playing. It doesn't matter how high the expected value is if a 1/2^100 probability event is unlikely to happen in the entire lifetime of the Universe.
Suppose that at the beginning of the game, you decide to play no more than N turns. If you lose all your money by then, oh well; if you don't, you call it a day and go home.
So the longer you decide to play, the higher your expected value is. But is a 1/2^100 chance of winning 3^100 dollars really better than a 1/2 chance of winning 3 dollars? Just because the expected value is higher, doesn't mean that you should keep playing. It doesn't matter how high the expected value is if a 1/2^100 probability event is unlikely to happen in the entire lifetime of the Universe.