  Let $G$ be a countable group that splits as a free product of groups of the
form $G=G_1\ast\dots\ast G_k\ast F_N$, where $F_N$ is a finitely generated free
group. We identify the closure of the outer space
$P\mathcal{O}(G,\{G_1,\dots,G_k\})$ for the axes topology with the space of
projective minimal, \emph{very small} $(G,\{G_1,\dots,G_k\})$-trees, i.e. trees
whose arc stabilizers are either trivial, or cyclic, closed under taking roots,
and not conjugate into any of the $G_i$'s, and whose tripod stabilizers are
trivial. Its topological dimension is equal to $3N+2k-4$, and the boundary has
dimension $3N+2k-5$. We also prove that any very small
$(G,\{G_1,\dots,G_k\})$-tree has at most $2N+2k-2$ orbits of branch points.
