The question is in the title:

> **Question:** Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the [120-cell](https://en.wikipedia.org/wiki/120-cell)?

I consider only *convex* polytopes (convex hull of finitely many points) that are full-dimensional (not contained in a proper subspace).
And I consider a polytope to be distinct from the 120-cell if it has a non-isomorphic face-lattice.

It is known that any 4-polytope must have a 3-gonal, 4-gonal or 5-gonal face of dimension two.
The 120-cell has only 5-gonal faces of dimension two.