Convection with asymptotically large viscosity contrasts occurs in the stagnant lid regime characterized by the formation of an immobile lid on top of a convective layer. Convection beneath the lid is driven by the rheological temperature scale rather than the entire temperature drop across the layer. A boundary layer treatment of the mean field hydrodynamic equations with infinite Prandtl number, no-slip boundary conditions, and temperature and stress-dependent viscosity yields a scaling relationship for the stagnant lid heat flux. When the wavenumber is chosen such that the heat flux is maximized, the scaling relationship is in reasonable agreement with results of time-dependent, two-dimensional, convection simulations and boundary layer stability analysis. The best agreement with data occurs for pseudo-steady state, low Rayleigh number convection before the fully time-dependent asymptotic regime is reached.