To my knowledge, the "pushforward with compact supports" along $f \colon X \to Y$ for étale cohomology is such a thing. For topological spaces it is easy to define
$$f_!\mathcal{F}(U) = \{ s \in \mathcal{F}(f^{-1}U) \mid f|_{\operatorname{supp}(s)} \text{ is proper}\}$$
(you know, as the name says) but this relies on $U$ actually being a subset of $Y$, so for étale sheaves one simply chooses a factorization of $f$ as a composition $p \circ j$, where $p$ is proper and $j$ is an open immersion; for $p$, we define $p_! = p_*$, while for $j$ we take $j_!$ to be "extension by zero" which does work even in the étale topology; then $f_! = p_! j_!$.
There are many such factorizations (that is, many compactifications of $f$); that one even exists in general, under some finiteness hypotheses, is a theorem of Nagata, but as for vector space bases, there is no canonical one. And unlike for bases, there is also no "maximal object" in the class of functors obtained from these choices; they are all simply canonically isomorphic to each other.