Here’s an example where I ran into this recently.

Let’s say we play a “game”. Draw a random number A between 0 and 1 (uniform distribution). Now draw a second number B from the same distribution. If A > B, draw B again (A remains). What is the average number of draws required? (In other words, what is the average “win streak” for A?)

The answer is infinity. The reason is, some portion of the time A will be extremely high and take millions of draws to beat.