In real-life applications, which continuous distributions have NON-CONVERGENT expectations that require Lebesgue integration?

When computing expected values, Riemann integration works for only random variables with bounded support sets. For distributions with unbounded support sets, we can use improper Riemann integrals for "nice" cases like the normal or gamma distribution; these integrals do converge.

Theoretically, I know that there are "nasty" cases that require more complicated forms of integrals, because the Riemann integrals don’t converge. For these cases, we need other forms of integration, like Lebesgue integrals.

However, in real-life applications in statistics, are there any continuous distributions whose expectations involve "nasty" integrals that don’t converge?