All the graphs I want to discuss are finite, simple, and connected. A graph $G_1$ covers another graph $G_2$ if there is a surjective map $\pi : V(G_1) \to V(G_2)$ that sends edges to edges and such that for any $v \in V(G_2)$ and any preimage $w \in V(G_1)$ with $\pi(w) = v$, the map $\pi$ is a bijection when restricted to the vertices adjacent to $v$, $N(v)$, surjecting $N(w)$. The degree of $\pi$ is the size of $\pi^{-1}(w)$ for any $w \in V(G_2)$. Maybe more simply, there is a covering map $G_1 \to G_2$ as topological spaces.
Suppose $G$ is a 3-connected graph. Can $G$ have connected planar covers of arbitrarily large degree (as covers)? Such graphs would conjecturally embed on the the projective plane.
For a loop graph we sure can -- hence the connectivity assumption.
