  We use the polynomial formulation of the holomorphic anomaly equations
governing perturbative topological string theory to derive the free energies in
a scaling limit to all orders in perturbation theory for any Calabi-Yau
threefold. The partition function in this limit satisfies an Airy differential
equation in a rescaled topological string coupling. One of the two solutions of
this equation gives the perturbative expansion and the other solution provides
geometric hints of the non-perturbative structure of topological string theory.
Both solutions can be expanded naturally around strong coupling.
