Suppose $X_n$ are pairwise independent standard normal variables for $n \in \mathbb{N}$, and consider $S := \sup\{X_n\} \in \mathbb{R} \cup \infty$. What is the probability that $S$ is finite?
I can show $E(S)=\infty$ pretty easily, but that says nothing about the probability that $S<\infty$.
I'd really like the answer to be that $S<\infty$ almost surely, but I doubt that it's true for the normal distribution. If not, is there another distribution supported on all of $\mathbb{R}$ that has this property?