Claim: Great dirhombicosidodecahedron is the only degenerate uniform polyhedron with more than six faces meeting at a vertex, that has 124 faces, 24 edges, and 60 vertices.
Evidence: This is the only non-degenerate uniform polyhedron with more than six faces meeting at a vertex. 
It has 124 faces (40 triangles, 60 squares, and 24 pentagrams), 240 edges, and 60 vertices.