What polyhedra can be made using only concave pentagons and regular polygons?
Above is one example that I really like. It looks like two regular dodecahedra which crashed together. Its faces are 18 regular pentagons and 6 concave pentagons.
Here is a second nice example. It is like a prism with two regular 10-gon bases, but around the sides are 10 concave pentagons and 10 equilateral triangles. You can extend this, if you like, to make a larger polyhedron which looks like two icosidodecahedra which crashed.
For other interesting examples and a nice analysis, you can look up his paper:
Paul Gailiunas, "A Polyhedral Byway," Proceedings of Bridges 2001: Mathematical Connections in Art, Music, and Science, pp. 115-122.But work on it on your own first.