what is the simplest example of an etale cover which is not Galois? By "Galois morphism" I mean a morphism $f: Y \to X$ such that $Y \times_X Y$ is a disjoint union of schemes isomorphic to $Y$.
Let $X$ be reduced curve over afield of char. 0.
I wonder what is a simple example of an etale morphism $f: Y \to X$ which is not Galois in the above sense. 
update: What about a "geometric" example?  (i.e. the base field is algebraically closed).

You may take (over the characteristic zero field $\mathbb Q$)  the morphism $$f:Y=\mathbb A_{\mathbb Q(\sqrt [3]2)}^1 \to X=\mathbb A_{\mathbb Q}^1$$       corresponding to the inclusion $\mathbb Q[T] \hookrightarrow  \mathbb Q(\sqrt [3]2)[T]$   
Edit 
At Dima's request, here is a completely explicit geometric example. 
Let $A=\frac {\mathbb C[T,Z]}{(Z^3-3Z+2T)} $  and $g:Spec (A)\to \mathbb A^1_\mathbb C$ the étale morphism corresponding to the inclusion $\mathbb C[T] \hookrightarrow  A$    
Then $g$ is  not Galois because the field extension $\mathbb C(T)\hookrightarrow  K=Frac (A)=\mathbb C(T)(z) \;$ is not Galois since  the discriminant of $ \; Z^3-3Z+2T\;$ is $\;108(1-T^2)\;$ and it is not a square in $\mathbb C(T)$ .
However $g$ is not étale either since it is ramified over $T=\pm1$.
 The remedy is simple: delete these two points i.e. corestrict $g$ to $$X=A^1_\mathbb C\setminus \lbrace 1,-1\rbrace=Spec(\mathbb C[T,\frac {1}{T^2-1}])$$  and restrict $g$ to 
$$Y=g^{-1}(X)=Spec(B)$$
where $B=A[\frac {1}{T^2-1}]=\mathbb C[T,\frac {1}{T^2-1}](z)$.
Finally the required étale non Galois morphism is the restriction
$$f=res(g):Y\to X                  $$  
A final remark
Actually there exist  no non-trivial étale covers of the affine line $\mathbb A^1_k$ over an algebraically closed  field $k$ of characteristic zero . (The affine line is then said, unsurprisingly,  to be simply connected) .     
However in characteristic $p\gt 0$, surprisingly, there exist many étale covers of the affine line .
The simplest is the Artin-Schreyer cover $\; Spec \frac {\mathbb k[T,Z]}{(Z^p-Z-T)}\to \mathbb A^1_k=Spec (k[T]) \;$but there are loads more: this is still a topic of current research.
