Abstract The synchrotron cooling of relativistic electrons is one of the most effective radiation mechanisms in astrophysics. It not only accompanies the process of particle acceleration, but also has feedback on the formation of the energy distribution of the parent electrons. The radiative cooling time of electrons decreases with energy as t _SYN ∝ 1/E; correspondingly, the overall radiation efficiency increases with energy. However, this effect strictly limits the maximum energy of individual photons. Even in the so-called extreme accelerators, where the acceleration proceeds at the highest possible rate, , allowed in an ideal magnetohydrodynamic plasma, the synchrotron radiation cannot extend well beyond the characteristic energy determined by the electron mass and the fine-structure constant: . In this paper, we propose a model in which the formation of synchrotron radiation takes place in compact magnetic blobs located inside the particle accelerator and develop a formalism for calculations of synchrotron radiation emerging from such systems. We demonstrate that for certain combinations of parameters characterizing the accelerator and the magnetic blobs, the synchrotron radiation can extend beyond this limit by a several orders of magnitude. This scenario requires a weak magnetization of the particle accelerator, and an efficient conversion of gas internal energy into magnetic energy in sufficiently small blobs. The required size of the blobs is constrained by the magnetic mirroring effect, which can prevent particle penetration into the regions of strong magnetic field under certain conditions.