If $(X_n)_{n\geq0}$ is a martingale and $\tau$ is a stopping time satisfying some mild conditions, then the optional stopping theorem says that
$$\mathbb E[X_\tau]=X_0.\tag{1}$$
Thus, if you're playing a fair game of chance, then the strategy that you use to exit the game has no effect on your expected earnings. There's no way to get smart and "beat the system" (or conversely, lose out because of a lousy strategy).
More generally, if $(X_n)_{n\geq0}$ is a submartingale or supermartingale, then we respectively have that
$$\mathbb E[X_\tau]\geq X_0\qquad\text{and}\qquad\mathbb E[X_\tau]\leq X_0.\tag{2}$$
Therefore, there's no way for a casino or player to beat a game that is not in their favor with a clever stopping strategy.
All this being said, there's an interesting difference between the statements in equations $(1)$ and $(2)$ above, which I've never really given any thought to until now:
Remark. On the one hand, $(1)$ tells you exactly what $\mathbb E[X_\tau]$ is when you have a martingale; it's just the initial condition $X_0$. In particular, all exit strategies in martingales are equally good/bad. On the other hand, $(2)$ leaves open the possibility that different exit strategies could yields better/worse outcomes for $\mathbb E[X_\tau]$, since we just have inequalities.
With this in mind, my (admittedly very open-ended) question is as follows:
Question. When you are given a submartingale or supermartingale: Is it possible to characterize (under any interesting set of assumptions) all the possible values that $\mathbb E[X_\tau]$ can take?
E.g., for a submartingale, $\mathbb E[X_\tau]\in[X_0,\infty)$; given $x\in(X_0,\infty)$, when does there exist a stopping time $\tau(x)$ such that $\mathbb E[X_{\tau(x)}]=x$? Obviously this can't exist for all submartingales, since every martingale is a submartingale and in this case $X_0$ is the only possibility. However, I can imagine that there might be some cases where we can get genuinely different outcomes depending on the stopping time. I've tried looking for such results in the literature, but have come up empty handed.