Building on recent progress in the study of the resurgent structure of theories based on quantum curves, I will discuss how the machinery of resurgence can be applied to asymptotic series that arise naturally as perturbative expansions in a strongly-coupled limit of topological string theory on a toric Calabi-Yau threefold. These asymptotic series show infinite towers of singularities in their Borel plane. The corresponding infinitely-many rational Stokes constants can be regarded as a conjectural class of topological invariants of the underlying theory, and they can be organized as coefficients of generating functions given by q-series. I will present an explicit analysis of a well-known example of toric Calabi-Yau geometry, whose resurgent structure turns out to be analytically solvable. This leads to proven exact formulae for the Stokes constants. This analytic approach is then straightforwardly extended to the dual weakly-coupled limit of topological strings on the same background, which has been studied numerically in a recent work by Gu and Mariño. This talk is based on arXiv:2212.10606.