A 1-D global average ionosphere and thermosphere (GAIT) model is used to examine the climatological behavior of the upper atmosphere, subject to both extremely low and high solar flux. These extremes are justified, in part, by the Maunder Minimum and Grand Maximum epochs described by J. A. Eddy, as well as other studies involving cosmogenic isotopes and Sun-like stars. As the irradiance falls below normal solar minimum levels, the concentration of O+ decreases rapidly relative to the molecular ions, such that the ratio foF2/foF1 approaches unity. When subject to exceptionally high solar fluxes, the ionospheric peak electron density (NmF2) unexpectedly plateaus, remaining relatively constant even as the photon flux continues to increase. In both cases, the state of the underlying thermosphere, particularly the neutral gas temperature, is found to be largely responsible. Model trends are discussed in relation to ionospheric observations, specifically the preponderance of so-called ionospheric G conditions at solar minimum and foF2 saturation at solar maximum, as well as the problem of Earth's global helium budget.