Heatwaves are intensifying at a fast pace, and how much further they can strengthen is unknown. Our study seeks to estimate a physical upper limit to surface air temperature. We show that, unlike what recent work suggested, the intensity of the most extreme heatwaves may not be constrained by the onset of thunderstorms. Instead, the limit is set by the development of a several-kilometer-deep layer of well-mixed air above the ground, and modulated by a very hot and unstable near-surface layer.