A small volume of space, nearly on-axis behind a gravitational lens with respect to a given source, will receive a greatly increased radiation flux. In the idealized case of a point mass lens acting on a point source in complete isolation, the volume will approach zero only as the flux tends to infinity; in fact, the volume weighted rms flux is divergent. In realistic cases, finite source size and the effects of other gravitational deflections (i.e., non-zero shear) limit the maximum flux and considerably complicate the physics, but very large fluxes are still produced in small volumes. We consider the physics and statistics of these Extreme Gravitational Lensing Events (EGLE) and present an initial examination of their possible astrophysical effects for various known and putative populations of lensing objects and sources, with particular attention to the case in which finite source size is important but shear is not.