This system provides perhaps one of the simplest possible models of moist convection and yet, because it is highly nonlinear, its behavior is complex and puzzling. The controlling nondimensional parameters include a conventional Rayleigh number, a moist Rayleigh number related to the latent heat of condensation, a parameter determining the relationship of temperature to buoyancy and a parameter in a Clausius--Clapeyron like equation. We find a 'drizzle' solution analytically, which the numerical model reproduces at low Rayleigh number, but at higher Rayleigh number the model produces active convection in regimes where the corresponding dry model is stable. In the convecting regime we look at how the vertical distribution of various fields (e.g., relative humidity, buoyancy), as well has the vertical heat flux, vary with nondimensional parameters. We find that some fields appear to tend toward a limit that is largely independent of Rayleigh number. However, in general, the heat and moisture fluxes do not go to a limit that is independent of Rayleigh number (or molecular viscosity). We discuss if and how this system is relevant, or not, to real atmospheric convection.