As I understand your question, and electron (or a hole) in a doped semiconductor meets your requirements; there are an (effectively) infinite number of energy levels available to these particles between the Fermi sea and the binding energy (work function) of the semi-conductor.

As I understand your question, an electron (or a hole) in a doped semiconductor meets your requirements; there are an (effectively) infinite number of energy levels available to these particles between the Fermi sea and the binding energy (work function) of the semi-conductor.