Why only finite morphisms in etale fundamental group? 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.

 A: As mentioned in other comments, there is a "pro-étale fundamental group" considered by Bhatt and Scholze. It is introduced in Chapter 7. of their article "The pro-étale topology for schemes". It is a topological group that is a so-called "Noohi group". For a connected (locally topologically noetherian) scheme $X$, it parameterizes the schemes $Y \rightarrow X$ that are étale and satisfy the valuative criterion of properness - the authors call such $Y$ "geometric covers" of $X$. Because we do not assume $Y$ to be of finite type over $X$, the map is not necessarily proper and we get more than just finite covers.
Another words, there is an equivalence of categories between the category of (possibly infinite) discrete sets with a continuous action of $\pi_1^{\mathrm{pro\acute{e}t}}(X)$ and the category of geometric covers of $X$.
The pro-\'etale fundamental group generalizes both the usual étale fundamental group and the "SGA3 fundamental group". The group $\pi_1^{\mathrm{\acute{e}t}}(X)$ is the profinite completion of $\pi_1^{\mathrm{pro\acute{e}t}}(X)$ and $\pi_1^{\mathrm{SGA3}}(X)$ is the pro-discrete completion of $\pi_1^{\mathrm{pro\acute{e}t}}(X)$. The situation is as follows:
If the scheme is normal, all three groups match, i.e. every geometric cover is finite. In the case of the nodal curve there exists an infinite geometric cover. In that case one can show $\pi_1^{\mathrm{pro\acute{e}t}}(X)=\pi_1^{\mathrm{SGA3}}(X)=\mathbb{Z}$  (assuming the base field was algebraically closed). However, for more complicated non-normal schemes $\pi_1^{\mathrm{pro\acute{e}t}}(X)$ gives more than $\pi_1^{\mathrm{SGA3}}(X)$: see Example 7.4.9. of "The pro-étale topology for schemes".
