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1. Knot polynomials like the Jones polynomial
2. Perturbative expansions of Feynman Integrals
3. Heat kernel asymptotics, and other universal polynomials in characteristic classes.
4. Generating functions associated to combinatorial problems.
5. Poincare Polynomials of Topological Spaces.
6. Hilbert Polynomials.
7. Certain families of orthogonal polynomials, generally associated to representation theory.
8. The A-polynomial

I am sure there are more.