I've seen at least a couple of sources (e.g. the nLab) that do not require the covering families in a coverage to be closed under pull-backs. The question is simple: are there any interesting examples that justify such generality?
Elaboration
Of course, nothing is wrong with greater generality per se, but in this case, it seems as if the greater generality comes at a cost. Perhaps most obvious is that it makes the definition of a coverage more complicated:
A coverage on a category is an assignment to each object $U$ a collection of precovers, called the covers of $U$, that has the property that, for every cover of an object $U$ and map from $V$ to $U$ there is a cover of $V$ that has the property that every element in the cover of $V$ composed with the map into $U$ factors through an element in the cover of $U$.
in contrast to
A coverage is an assignment to each object $U$ a collection of precovers, the covers of $U$, that has the property that a pull-back of a cover is a cover.
(Here, the term precover on $U$ means just a collection of maps with codomain $U$.)
Arguably more significant is that, if our coverage is not closed under pull-backs, then the usual "equalizer definition" of sheaves no longer makes sense. Instead, one must phrase things in terms of compatible collections of sections. This, however, has the unfortunate property that it does not make sense for any codomain category whose objects don't have underlying sets. On the other hand, using the "equalizer definition", we can define sheaves $\mathcal{O}\colon (\mathbf{X},\widetilde{\mathcal{U}})\rightarrow \mathbf{S}$ for any category $\mathbf{S}$: in this sense, it seems as if in asking for greater generality for sites we have lost greater generality for sheaves.
Thus, if one really insists upon allowing coverages not closed under pull-backs, this suggests another question: in this more general context, is there a convenient way to rephrase the sheaf axiom so as to make sense for arbitrary codomain category?