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