  Consider a random walk with a drift to the right on $\{0,\ldots,k\}$ where
$k$ is random and geometrically distributed. We show that the tail $P[T>t]$ of
the length $T$ of an excursion from $0$ decreases up to constants like
$t^{-\varrho}$ for some $\varrho>0$ but is not regularly varying. We compute
the oscillations of $t^\varrho\,P[T>t]$ as $t\to\infty$ explicitly.
