The following (or at least a good approximation of it) might be well-known, but I could not find it explicitly stated:
Given a (symmetric) random walk on $\mathbb{Z}$ starting at 0. Let $M$ be the largest [absolute] value it took before reaching 0 again. $M$ can clearly take arbitrarily large values, but is the expected value of $M$ infinite?
To make this a bit more formal: let $X_i$ be i.i.d. uniformly distributed on $\lbrace \pm 1 \rbrace$ and consider $S_i = \sum_{i=0}^j X_i$. Let $T = \min \lbrace t \geq 1 \mid S_t =0 \rbrace$. Then $M = \max \lbrace |S_j| \mid j \leq T \rbrace$.
Since the expected value of $T$ is infinite, this suggests that the same could hold for $M$.
An excursion away from 0 starts with step to $1$ or a step to $-1$. By the evident symmetry, it's enough to consider the first case. For the r.w. started at $1$, the chance to hit $n$ before $0$ is well known to be $1/n$. Thus, $$ P[M\ge n] = 1/n $$ and indeed $E[M]=+\infty$.