  We define for each g>=2 and k>=0 a set M_{g,k} of orientable hyperbolic
3-manifolds with $k$ toric cusps and a connected totally geodesic boundary of
genus g. Manifolds in M_{g,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 M_{g,k} for a fixed k has growth type g^g. We completely
describe the non-hyperbolic Dehn fillings of each M in M_{g,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 boundary-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 boundary-reducible fillings are handlebodies.
