Let G be a minor-closed graph class. We say that a graph G is a k-apex of G if G contains a set S of at most k vertices such that G ∖ S belongs to G. We denote by A k ( G ) the set of all graphs that are k -apices of G. We prove that every graph in the obstruction set of A k ( G ), i.e., the minor-minimal set of graphs not belonging to A k ( G ), has order at most 2 2 2 2 poly ( k ), where poly is a polynomial function whose degree depends on the order of the minor-obstructions of G. This bound drops to 2 2 poly ( k ) when G excludes some apex graph as a minor.