Dehn filling of cusped hyperbolic 3-manifolds with geodesic boundary

We define for each g >= 2 and k >= 0 a set Mg,k of orientable hyperbolic 3-manifolds with k toric cusps and a connected totally geodesic boundary of genus g. Manifolds in Mg,k have Matveev complexity g + k and Heegaard genus g +1, and their homology, volume, and Turaev-Viro invariants depend only on g and k. In addition, they do not contain closed essential surfaces. The cardinality of Mg,k for a fixed k has growth type g^g. We completely describe the non-hyperbolic Dehn fillings of each M in Mg,k , showing that, on any cusp of any hyperbolic manifold obtained by partially filling M , there are precisely 6 non-hyperbolic Dehn fillings: three contain essential discs, and the other three contain essential annuli. This gives an infinite class of large hyperbolic manifolds (in the sense of Wu) with ∂ -reducible and annular Dehn fillings having distance 2, and allows us to prove that the corresponding upper bound found by Wu is sharp. If M has one cusp only, the three ∂ -reducible fillings are handlebodies.