The Teichmuller space of an orientable surface S has a proper action of the mapping class group Mod(S), with quotient the moduli space M of Riemann surfaces. To an algebraic topologist, the quotient M is of interest because it serves as a rational K(pi,1) for Mod(S), but to an algebraic geometer it is unsatisfactory because it is not a projective variety, and does not parameterize families of surfaces which might degenerate. Deligne and Mumford compactified M by adding surfaces with nodal singularities, and Bers described a lift of this to a partial compactification of Teichmuller space, whose combinatorics are given by the curve complex C(S). Algebraic geometers then defined a ``tropical” version of M, which lifts in a natural way to a cone on C(S), so that this cone might therefore be considered the “tropicalization” of Teichmuller space. I will use handlebodies to describe a space intermediate between Teichmuller space and M whose “tropicalization” in this sense is a cone on the simplicial closure of Outer space. If S has no marked points this intermediate space can be identified with Schottky space. This is joint work with R. Ramadas, R. Silversmith and R. Winarski.