Say that we have the following expression:
$$E_0(\max(0,Y_1))$$ Where $E_t$ denotes the expecation at time $t$, conditional on $\{Y_0,...,Y_t\}$. i.e. $E_t(Y_{t+1})=E(Y_{t+1}|Y_0,...,Y_t)$ But now we iterate it, taking the expectation of time $0$ of the maximum of $0$ and our expectation at time $1$: $$E_0(\max(0,E_1(\max(0,Y_2)))$$ And again: $$E_0(\max(0,E_1(\max(0,E_2(\max(0,Y_3))))$$ And $T$ times: $$E_0(\max(0,E_1(\max(0,...E_{T-1}(\max(0,Y_T)...))))$$
If the $\max$ operators weren't there, then we would simply apply the law of iterated expectations to get $E_0(Y_T)$. But we cannot do this obviously, because of the max functions.
Is this even remotely tractable? You can assume anything you want about the distribution of $Y_t$, whatever is needed to make it tractable. I have assumed the normal distribution, but did not get even beyond a single iteration. (I'm ok with approximations). I actually need $Y_t$ to have not-necessarily-zero mean, but I suspect that this will also make it less tractable.