Suppose we have the following game that's free to play. The game consists of an infinite number of stages, where in each stage you must bet your entire bankroll to quadruple it with probability $3/4$ and lose it with probability $1/4$. You may stop playing the game anytime you wish and collect your winnings, and you are given 1 unit for free to play with in the first stage.
My question is: what's the best strategy for when to stop? Each stage of the game has positive expected value when considered alone, as we win $4b$ with probability $3/4$ and lose $b$ with probability $1/4$, so that the EV is $(11/4)b$, where $b$ is our bankroll. However, in the long run we will lose almost surely, and so the "correct" strategy can't be to just play whenever playing is positive expected value.
I've considered "double or nothing" strategies, where we fix a value $n$ and stop playing only when we have earned at least $n$. This gives us a new random variable $X_n$, whose expected value will be
$$n\left(\frac{3}{4}\right)^{\log_4(n)}$$
as we reach our desired level after $\log_4(n)$ rounds of quadrupling and earn $n$. However, this goes to infinity, and so again it seems that the "optimal" strategy in terms of expected value is to keep going for double or nothing until we bust.
I'm sure this has a simple resolution (or maybe it's a special case, since this theoretical game is far from fair). Given this game, when should we stop playing? Thanks in advance!