Polymers with no molecular mass distribution?
Living polymerizations can afford nearly zero polymer polydispersivity.  You can go anion (Kraton thermoplastic elastomers - and tell us about the one silicon atom in each polymer molecule), free radical (TEMPO or PROXYL capping of the free radical end, thermally activated), redox (copper halide), olefin metathesis polymerization (ROMP and ADMET), and other clevernesses.

Proteins are made from a DNA template via RNA, one AA residue at a time. There is no dispersivity.  Post-synthesis modification, e.g., glycosylation in erythropoietin, can be polydisperse.