Compact 3-manifold with compressible boundary tori? There is much to be said about 3-manifolds with boundary consisting of a possibly empty collection of incompressible tori.
I don't seem to know where to look to find much about 3-manifolds with compressible torus boundary.
If $M^3$ is compact, with infinite $\pi_1$, and with non-empty boundary, consisting only of compressible tori, a solid torus is an option, but what else can happen?

Assuming that your manifold is, say, compact, connected and orientable, whose boundary is a union of tori, then $M$ is homeomorphic to a connected sum of a manifold $M'$ whose boundary tori are all incompressible and some number of solid tori. This is a consequence of the Loop Theorem.
