- Knot polynomials like the Jones polynomial
- Perturbative expansions of Feynman Integrals
- Heat kernel asymptotics, and other universal polynomials in characteristic classes.
- Generating functions associated to combinatorial problems.
- Poincare Polynomials of Topological Spaces.
- Hilbert Polynomials.
- Certain families of orthogonal polynomials, generally associated to representation theory.
- The A-polynomial
I am sure there are more.