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There are the Platonic Solid (regular polyhedra)

http://mathworld.wolfram.com/PlatonicSolid.html

The most spherical is probably the duodecahedron (12 pentagons)

Then there are the Archimedean Solids (semi regular)

http://mathworld.wolfram.com/ArchimedeanSolid.html

The most spherical is probably the soccer ball, but it uses two different polygons.