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We compute arbitrary-order asymptotic expansions of standing water waves, generalizing the conformal mapping framework of Schwartz and Whitney (1981) to handle finite-depth domains and/or surface tension effects. The calculations are performed with up to 900 digits of precision using MPFR and parallelized to run on a supercomputer. We explore Pade approximations of the expansions and observe branch cuts that leave gaps in the families of solutions. The Pade approximants agree closely with solutions computed by a shooting method on both sides of each gap. Bivariate rational approximations allow us to study the solutions near resonant depths where the expansion breaks down. We also explore analytic continuation of standing waves into the upper half-plane and observe branch cuts there as well.