An open 3-manifold is *tame* if it is homeomorphic to the interior of a compact manifold. Is there a (known) example of an open 3-manifold that is not tame, has finitely generated fundamental group and universal cover homeomorphic to $\mathbb R^3$?



Non-tame 3-manifolds covered by the Euclidean space
Yes, see the following paper of Freedman and Gabai for lots of examples:


Freedman and Gabai, Covering a nontaming knot by the unlink. Algebr. Geom. Topol. 7 (2007), 1561–1578. 


