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