The yellow-and-red polyhedron in the compound below is the truncated icosahedron, one of the Archimedean solids. The blue figure is its dual, the pentakis dodecahedron, which is one of the Catalan solids.

The next image shows the convex hull of this base/dual compound. Its faces are kites and rhombi.

Shown next is the dual of this convex hull, which features regular hexagons, regular pentagons, and isosceles triangles.

Next, here is the compound of the last two polyhedra shown.

Continuing this process, here is the convex hull of the compound shown immediately above.

This latest convex hull has an interesting dual, which is shown below. It blends characteristics of several Archimedean solids, including the rhombicosidodecahedron, the truncated icosahedron, and the great rhombicosidodecahedron.

This process could be continued indefinitely — making a compound of the last two polyhedra shown, then forming its convex hull, then creating that convex hull’s dual, and so on.

All these polyhedra were made using Stella 4d: Polyhedron Navigator, which you can purchase (or try for free) at http://www.software3d.com/Stella.php.