Severe convection, responsible for hazards such as tornadoes, flash floods, and hail, is usually preceded by abundant Convective Available Potential Energy (CAPE). In this work, we use a Lagrangian approach to study how anomalously large values of CAPE build up in various regions. Nearly all extreme values of CAPE arise from surface fluxes underneath a capping inversion over several diurnal cycles, but the origin of the capping inversion and the diurnal cycle of surface fluxes differ around the world. In some regions, such as North America and Europe, the air above the boundary layer must be much warmer than usual to form this capping inversion, while in other regions, especially the Middle East and Central Africa, a capping inversion is common. Additionally, high CAPE occurrences that are over land (those in the Americas, Europe, Africa, and East Asia) tend to lose their capping inversions before the time of maximum CAPE due to large diurnal cycles of sensible heating, while those that occur over coastal waters (in the Middle East, Northern Australia, South Asia, and the Mediterranean) usually retain a substantial capping inversion. Uniquely, CAPE in Southeast Australia often builds up due to cooling aloft rather than to boundary layer warming. These results show that one hoping to understand or predict CAPE patterns must understand a variety of mechanisms acting in different regions.