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.A&A proofs: manuscript no. main of it), as well as the observation of the relatively abundant low frequency LIGO events. In this work we estimate the probability of observing, not only luminous stars at z > 1 but also discuss other very compact but intrinsically energetic phenomena, such as SNe, or GW.In order to observe distant gravitationally lensed objects, one needs large magnification factors that compensate for the increase in luminosity distance. The smaller probability of lensing is partially compensated by the larger volume which, between z ≈ 0.3 and z ≈ 1.3, grows approximately as (1 + z) 3 (and at a slower rate at higher redshifts). More precisely, the volume per redshift interval peaks at z ≈ 2.5 (with about 55 Gpc 3 in a redshift interval of thickness ∆z = 0.1). Regarding the probability of lensing, background objects at high redshifts have a higher probability of intersecting a gravitational lens along the line of sight. This probability is described by the optical depth. It is well known that the optical depth grows rapidly between z=0 and z=1. Between z=1 and z=3 it continues to grow although at a much slower pace. Beyond z =3 -5, the optical depth is still growing but much more slowly and beyond z =5 it only grows by percent values, specially if one considers large magnification factors where the presence of caustics is required (these caustics are expected to be rare for lenses beyond z ≈ 3). For sources (or events) that trace the star formation history, and at redshifts of the background source between 1 and 3, the small probability of magnification factors can be compensated by the larger volume element and increased volumetric density. In this redshift interval, the probability of seeing extremely magnified events can be maximum.Examples of extreme magnification are the aforementioned ...