I'm somewhat out of my comfort zone here, but I think this is right.

The figure-8 knot complement $M_8$ is *universal*, meaning that every closed 3-manifold arises as a Dehn filling on a finite-sheeted covering space of $M_8$.  So the family $\lbrace M_8, D^2\times S^1\rbrace$ generates all 3-manifolds.

I don't have time to look at the references right now, but I'll try to get back to it later.