  Let Map_T(K,X) denote the mapping space of continuous based functions between
two based spaces K and X. If K is a fixed finite complex, Greg Arone has
recently given an explicit model for the Goodwillie tower of the functor
sending a space X to the suspension spectrum \Sigma^\infty Map_T(K,X). Applying
a generalized homology theory h_* to this tower yields a spectral sequence, and
this will converge strongly to h_*(Map_T(K,X)) under suitable conditions, e.g.
if h_* is connective and X is at least dim K connected. Even when the
convergence is more problematic, it appears the spectral sequence can still
shed considerable light on h_*(Map_T(K,X)). Similar comments hold when a
cohomology theory is applied. In this paper we study how various important
natural constructions on mapping spaces induce extra structure on the towers.
This leads to useful interesting additional structure in the associated
spectral sequences. For example, the diagonal on Map_T(K,X) induces a
`diagonal' on the associated tower. After applying any cohomology theory with
products h^*, the resulting spectral sequence is then a spectral sequence of
differential graded algebras. The product on the E_\infty -term corresponds to
the cup product in h^*(Map_T(K,X)) in the usual way, and the product on the
E_1-term is described in terms of group theoretic transfers. We use explicit
equivariant S-duality maps to show that, when K is the sphere S^n, our
constructions at the fiber level have descriptions in terms of the
Boardman-Vogt little n-cubes spaces. We are then able to identify, in a
computationally useful way, the Goodwillie tower of the functor from spectra to
spectra sending a spectrum X to \Sigma ^\infty \Omega ^\infty X.
