Motivated by the asymptotic theory of the almost-highest traveling water wave, we develop a family of extremely non-uniform (but smoothly defined) meshes and show how to compute quantities such as the Hilbert transform with spectral accuracy in order to solve a variant of the Nekrasov formulation of the traveling water wave problem very close to the limiting Stokes wave. We also consider time-dependent applications in population genetics with singular boundary conditions, and, if time permits, singularity formation in 4-dimensional Ricci flow.