A sphere has no edges, no faces (at least, no flat ones), and no vertices.
A torus (doughnut) has no edges, no faces, and no vertices.
There are probably other curved solids that do this.
But, if you mean polyhedra (made of only flat surfaces), I don't think there are any.
Euler's Formula. V - E + F = 2
If V = E = F, then V - V + V = 2, then V = 2, E = 2, F = 2. There are no solid polyhedra with only 2 faces.