Etale covers of the affine line
In characteristic p there are nontrivial etale covers of the affine line, such as those obtained by adjoining solutions to x^2 + x + f(t) = 0 for f(t) in k[t]. Using an etale cohomology computation with the Artin-Schreier sequence I believe you can show that, at least, the abelianization of the absolute Galois group is terrible.
What is known about the absolute Galois group of the affine line in characteristic p? In addition, can spaces which are not A^1 (or extensions of A^1 to a larger base field) occur as covers?