Graphs with multiple edge costs arise naturally in the route planning domain when apart from travel time other criteria like fuel consumption or positive height difference are also objectives to be minimized. In such a scenario, this paper investigates the number of extreme shortest paths between a given source-target pair s, t. We show that for a fixed but arbitrary number of cost types d ≥ 1 the number of extreme shortest paths is in n^O(log^{d-1}n) in graphs G with n nodes. This is a generalization of known upper bounds for d = 2 and d = 3.