Conformal branched cover from the hyperbolic plane to the euclidean plane The euclidean plane can conformally branch cover the sphere. This is witnessed by the tiled Peirce quincuncial projection. 

Yes, there is. Here is a geometric construction (which could easily be visualized if one wants to do so): Let $S$ be a square (i.e., a quadrilateral with 4 equal sides and 4 equal angles) in the hyperbolic plane with angles of $\pi/4$, and let $f:S \to [0,1]^2$ be the conformal map of $S$ to the unit square, mapping vertices to vertices. Now the map $f$ can be extended by reflection to a map of the hyperbolic plane onto the Euclidean plane. The resulting map will have branch points of degree two over all points in the integer lattice, since it is angle-doubling at those points.