I have some indipendent, discrete variables $X_1$ through $X_n$, which don't necessarily share the same (known) probability distributions or possible values. What is the expected value of $\max\left\{X_1, X_2,\dots, X_n\right\}$?
This question is related but not quite general enough (though I'm sure this could be adapted).
Also, this question comes from a generalization of what I did to answer this code golf question which is about when all variables are dice rolls, for which I found the closed form $6 - \sum_{i=1}^5\left(\frac i 6\right)^n$ (not finding the question I linked above!) and explained it this way which I definitely have no idea how to generalize.
If $X_i>0$ and $F_i(x)=\Pr(X_i\leq x)$ then $\Pr(\max X_i\leq x)=F_1(x)\ldots F_n(x)$ and $E(\max X_i)=\int_0^{\infty}(1-F_1(x)\ldots F_n(x))dx.$