Expected hitting time of +1 for a fair walk

A particle starts at $0$ on the integers and at each step moves up or down by one with equal probability $\tfrac12$. What is the expected number of steps until it first reaches position $1$? If this expectation is infinite, answer $-1$.

$-1$ (the expectation is infinite). Conditioning on the first step gives $m=1+\tfrac12\cdot 2m$, i.e. $m=1+m$, which has no finite solution. The walk reaches $1$ with probability one, but the hitting time has a heavy tail ($P(T>n)\sim c/\sqrt n$), so its mean diverges.