  Extreme magnifications of distant objects by factors of several thousand have
recently become a reality. Small very luminous compact objects, such as
supernovae (SNe), giant stars at z=1-2, Pop III stars at z>7 and even
gravitational waves from merging binary black holes near caustics of
gravitational lenses can be magnified to many thousands or even tens of
thousands thanks to their small size. We explore the probability of such
extreme magnifications in a cosmological context including also the effect of
microlenses near critical curves. We show how a natural limit to the maximum
magnification appears due to the presence of microlenses near critical curves.
We use a combination of state of the art halo mass functions, high-resolution
analytical models for the density profiles and inverse ray tracing to estimate
the probability of magnification near caustics. We estimate the rate of
highly-magnified events in the case of SNe, GW and very luminous stars
including Pop III stars. Our findings reveal that future observations will
increase the number of events at extreme magnifications opening the door not
only to study individual sources at cosmic distances but also to constrain
compact dark matter candidates.
