I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.


It is well known that the étale fundamental group in positive characteristics is a very difficult object, especially so in the non-projective case due to possibly wild ramification at infinity. I'm not even sure if there are examples of the kind above. Is this known?



Simply connected quasi-projective varieties in positive characteristic
This is an answer to Pete's question on simply connected affine varieties (I can not put it in a comment because of space limitation). 


I think that in positive characteristic $p$, no affine **irreducible** variety $X$ of positive dimension is simply connected. We can assume $X=\operatorname{Spec}(A)$ integral because $\pi\_1$ is insensible to nipotent elements (SGA IX.4.10). Let $k[t\_1,\ldots, t\_d] \subseteq A$ be a finite extension with minimal degree $[k(A):k(t\_1,\ldots, t\_d)]$. Consider the étale cover $Y\to \mathbb A^d\_k= \operatorname{Spec}(k[t\_1,\ldots, t\_d])$ defined by $s^p-s=t\_1$. Then $X\times\_{\mathbb A^d\_k} Y\to X$ is an étale cover of degree $p$. As $k(Y)$ and $k(X)$ are linearly disjoint over $k(t\_1,\ldots, t\_d)$ ($k(Y)$ is Galois over $k({\bf t}):=k(t\_1,\ldots, t\_d)$ and $k(Y)\cap k(X)=k({\bf t})$), the tensor product $k(Y)\otimes\_{k({\bf t})} k(X)$ is a field. This implies that $X\times\_{\mathbb A^d\_k} Y$ is connected. 


