  The motivic nearby fiber is an invariant obtained from degenerating a complex
variety over a disc. It specializes to the Euler characteristic of the original
variety but also contains information on the variation of Hodge structure
associated to the degeneration which is encoded as a limit mixed Hodge
structure. However, this invariant is difficult to compute in practice. Using
the techniques of tropical geometry we present a new formula for the motivic
nearby fiber. Moreover, since there is a range of available software
implementing the main algorithms in tropical geometry, our formula can be
computed in practice. We specialize to the case of families of sch\"{o}n
complex hypersurfaces of tori where we provide explicit formulas describing the
action of the unipotent part of monodromy on the graded pieces (with respect to
the Deligne weight filtration) of the cohomology with compact supports. These
families are described combinatorially by a polyhedral subdivision of the
associated Newton polytope. We develop new mixed Hodge theory-inspired
combinatorial invariants of such subdivisions, among them the `refined limit
mixed $h^*$-polynomial'. These invariants are related to Stanley's
combinatorial study of subdivisions: in a companion combinatorial paper whose
results are applied here, we situate our invariants in Stanley's theory where
they become multi-variable extensions of his invariants. Our results generalize
work of Danilov and Khovanski{\u\i} and Batyrev and Borisov on the Hodge
numbers of hypersurfaces. We also present analogous formulas describing the
action of the unipotent part of monodromy on the intersection cohomology groups
of a family of sch\"on hypersurfaces of a projective toric variety.
