Most rational asymptotic studies of non-rotating Rayleigh–Bénard convection and its cousins have been restricted to the linear or weakly nonlinear regime. An important exception occurs for large Rayleigh-number thermal convection at effectively infinite Prandtl number, i.e. fast but very viscous convection. In this scenario, the temperature field exhibits a layer-like structure surrounding an isothermal core and, crucially, the momentum equation linearizes. These features have been exploited by several authors to obtain semi-analytical nonlinear solutions. At O(1) Prandtl number, the fluid dynamics in the vortex core is dominated by nonlinear inertial rather than linear viscous effects, substantially altering the vortex structure. Here, it is shown that a combination of matched asymptotic analysis and global conservation constraints can be used to obtain a semi-analytic yet strongly nonlinear description of two related flows: (i) Rayleigh–Bénard convection between constant heat-flux boundaries at unit Prandtl number, and (ii) Langmuir circulation (LC), a wind and wave-driven convective flow commonly observed in natural water bodies. A simple analytical prediction is given for the roll-vortex amplitude, which is shown to be independent of the horizontal wavenumber of the convection pattern. In marked contrast to weakly nonlinear convection cells, the fully nonlinear asymptotic solutions exhibit flow features relevant to turbulent convection including the complete vertical redistribution of the basic-state temperature (or, for LC, downwind velocity) field. Comparisons with well-resolved pseudospectral numerical simulations of the full two-dimensional governing equations confirm the accuracy of the asymptotic results.