  A filling Dehn sphere $\Sigma$ in a closed 3-manifold $M$ is a sphere
transversely immersed in $M$ that defines a cell decomposition of $M$. Every
closed 3-manifold has a filling Dehn sphere. The Montesinos complexity of a
$3$-manifold $M$ is defined as the minimal number of triple points among all
the filling Dehn spheres of $M$. A sharp upper bound for the Montesinos
complexity of the connected sum of two 3-manifolds is given.
