In general, it is not uncommon in probability theory to encounter the result $\mathbb{E}[\tau]=\infty$ but $\mathbb{P}(\tau = \infty)=0$ for some stopping time $\tau$.
A specific example could be the answers to this question here, which show that the stopping time $\tau$, defined as follows:
$$\tau = \inf \{ t \ge 0: B_{t} \ge W_{t} + e^{-t} \}$$
is unbounded (i.e. $\mathbb{E}[\tau]$ doesn't converge) but is almost surely finite.
($B_t$ and $W_t$ are two uncorrelated Standard Brownian Motions)
In general, whenever we have $\mathbb{E}[\tau]=\infty$ but $\mathbb{P}(\tau = \infty)=0$ for some stopping time $\tau$, how do we interpret such a result in plain English?
If I had to explain what this result means to a 1st year undergrad, or even someone starting a graduate course in probability, how could I explain this intuitively? Would there be any real-world practical examples where such result would hold?
I.e. "how can something take on average infinite amount of time, but with certainty, it takes less than an infinite time".