  Theories of massive gravity with one or two dynamical metrics generically
lack stable and observationally-viable cosmological solutions that are
distinguishable from $\Lambda$CDM. We consider an extension to trimetric
gravity, with three interacting spin-2 fields which are not plagued by the
Boulware-Deser ghost. We systematically explore every combination with two free
parameters in search of background cosmologies that are competitive with
$\Lambda$CDM. For each case we determine whether the expansion history
satisfies viability criteria, and whether or not it contains
beyond-$\Lambda$CDM phenomenology. Among the many models we consider, there are
only three cases that seem to be both viable and distinguishable from standard
cosmology. One of the models has only one free parameter and displays a
crossing from above to below the phantom divide. The other two provide scaling
behavior, although they contain future singularities that need to be studied in
more detail. These models possess interesting features that make them
compelling targets for a full comparison to observations of both cosmological
expansion history and structure formation.
