When we have two different locally univalent circle packings of one triangulation of a polygon, we can interpret the simplicial map between the corresponding immersions of the triangulation as a discrete analogue of a conformal map. In a branched circle packing, the cycle of neighbours of a circle may wind around it more than once before closing up. Branched circle packings allow us to think about discrete analogues of analytic mappings that are not locally univalent. Thanks to work of Bowers and Stephenson the circle packing analogue of a Blaschke product is well understood. There are also some beautiful constructions of circle packing analogues of rational functions. We can think of a branched circle packing as a restricted kind of packing of disjoint metric balls in a Riemann surface that is equipped with a path metric pulled back from the spherical, Euclidean or hyperbolic metric under an analytic map. The restriction is this: if a closed ball in the packing contains a singular point of the metric then that singular point must be unique and it must be the centre of the ball. Removing this restriction leads to the notion of generalized branching that was introduced by Ashe, Crane and Stephenson. In the talk I want to share an ambitious idea for trying to prove a version of the Koebe-Andreev-Thurston theorem for generalized branched circle packings, by using the principle of invariance of domain and studying the degenerations that can occur when Oded Schramm’s amazing metric packing theorem is applied to a sequence of mollifications of the singular path metric.