We explore atmospheric escape from close-in exoplanets with the highest mass-loss rates. First, we locate the transition from stellar X-ray and UV-driven escape to rapid Roche lobe overflow, which occurs once the 10–100 nbar pressure level in the atmosphere reaches the Roche lobe. Planets enter this regime when the ratio of the substellar radius to the polar radius along the visible surface pressure level, which aligns with a surface of constant Roche potential, is X/Z≳ 1.2 for Jovian planets (Mp≳ 100 M⊕) and X/Z≳ 1.02 for sub-Jovian planets (M p≈ 10–100 M⊕). Around a Sun-like star, this regime applies to orbital periods of less than two days for planets with radii of about 3–14R⊕. Our results agree with the properties of known transiting planets and can explain parts of the sub-Jovian desert in the population of known exoplanets. Second, we present detailed numerical simulations of atmospheric escape from a planet like Uranus or Neptune orbiting close to a Sun-like star that support the results above and point to interesting qualitative differences between hot Jupiters and sub-Jovian planets. We find that hot Neptunes with solar-metallicity hydrogen and helium envelopes have relatively more extended upper atmospheres than typical hot Jupiters, with a lower ionization fraction and higher abundances of escaping molecules. This is consistent with existing ultraviolet transit observations of warm Neptunes, and it might provide a way to use future observations and models to distinguish solar-metallicity atmospheres from higher-metallicity atmospheres.