The partition function of topological string theory on a Calabi-Yau threefold is defined perturbatively as a sum over free energies of genus g Riemann surfaces.  I will review the construction of a differential ring of special functions on the moduli space of CY threefolds. I will show that the higher genus amplitudes can be expressed as polynomials in the generators of this ring, which significantly simplifies a Feynman diagram computation. The polynomial structure of the amplitudes can furthermore be exploited to determine specific monomials to all orders in perturbation theory. A universal Airy differential equation in the topological string coupling can be derived governing these monomials. Solutions of this equation also offer hints of non-perturbative topological strings.