Grothendieck's Galois theory without finiteness hypotheses

In "The pro-étale topology for schemes" (http://arxiv.org/abs/1309.1198), Bhatt and Scholze introduce the pro-étale fundamental group which seems to give a good answer to your question (see, e.g., Theorem 1.10). The pro-étale fundamental group also compares well against the usual étale fundamental group and the "SGA3 étale fundamental group".

Check out Section 2 of Noohi's paper Fundamental groups of topological stacks with slice property, Algebr. Geom. Topol. 8 (2008) pp 1333–1370, doi:10.2140/agt.2008.8.1333, arXiv:0710.2615.

Besides the references already given above, one can also mention §7 of SGA III, Exp.X (Caracterisation Et Classification Des Groupes De Type Multiplicatif, doi:10.1007/BFb0059008), where Grothendieck sketches the theory of an enlarged fundamental group ("groupe fondamental élargi").

This was later on formalized in terms of Galois toposes in Olivier Leroy's Phd "Groupoide Fondamental et Theoreme de van Kampen en Theorie des Topos" which unfortunately does not seem to be widely available. Roughly, it goes at follows. A topos $E$ is locally Galois if it is locally connected and if every object is a sum of locally constant objects: $SLC(E)=E$ (equivalently, if it is generated by its Galois objects [ a Galois object is a locally constant non-empty object that is a pseudo-torsor under its automorphism group]). Such a locally Galois topos can be recovered from the groupoid (in the categorical sense) of its points in the sense that the functor $E\rightarrow (Point(E))^\wedge$ defines an equivalence of toposes (here for a groupoid $C$, $C^\wedge$ denotes the topos of presheaves on $C$). To sum up, $E\mapsto Point(E)$ defines an equivalence between locally Galois toposes and groupoids. You can find more details in Vincent Zoonekynd's paper The fundamental group of an algebraic stack at this address http://zoonek.free.fr/Ecrits/

Then starting from a scheme $X$, you can consider the topos $SLC(\widetilde{X_{et}})$ of locally constant sheaves for the étale topology. This is a Galois topos, the automorphism group of a point is Grothendieck's enlarged fundamental group. If instead the étale topology you stick to the finite étale topology $X_{fet}$, where covers are given by surjective families of finite étale maps, your recover the traditional profinite fundamental group of SGA1.