Can one define a version of etale fundamental group which takes into account infinite etale covers? What properties of the usual etale fundamental group would fail for it?

P.S.: here one can find illuminating discussion:

As finite étale maps of complex varieties are equivalent to finite topological covering spaces, this definition begs the question: why have we restricted to finite covering spaces? There are at least two answers to this question, neither of which is new: the first is that the covering spaces of infinite degree may not be algebraic; it is the finite topological covering spaces of a complex analytic space corresponding to a variety that themselves correspond to varieties. The second is that Grothendieck’s étale $π_1$ classifies more than finite covers. It classifies inverse limits of finite étale covering spaces [SGA1, Exp. V.5, e.g., Prop. 5.2]. These inverse limits are the profinite-étale covering spaces we discuss in this paper (see Definition 2.3). Grothendieck’s enlarged fundamental group [SGA 3, Exp. X.6] even classifies some infinite covering spaces that are not profinite-étale.

locallyof finite presentation, so the only thing this would allow is certain rising unions (this does lead to a few interesting new covers, but not so many). A more interesting generalisation is to weakly étale morphisms, which leads to the pro-étale topology of Bhatt and Scholze (see also the relevant chapter in the stacks project). But you can never hope to have the exponential $\mathbb C\to\mathbb C^\times$ as a cover, because it is not algebraic. What is it that you're trying to do? $\endgroup$ – R. van Dobben de Bruyn Nov 25 '18 at 18:56