Typical extremely intense laser-matter interactions include ionization, plasma production and generation of secondary particles. Modern laser systems are able to generate short and intense laser pulses ionizing matter in the poorly explored barrier-suppression regime. The classical and quantum models of barrier-suppression ionization are proposed and a simple formula of the ionization rate both for the tunnel and the barrier-suppression regimes is derived. After ionization free electrons move in extremely intense laser field in radiation-dominated regime when the radiation losses strongly affect electron dynamics. We show that the electron trajectories in this regime become close to some asymptotic trajectories where the radiation losses are minimal. The particle velocity of the asymptotic trajectory is completely determined by the local and instant EM field. At high laser intensity the laser-matter interaction can be accompanied by the avalanche-like production of electron-positron plasma via QED cascading. We demonstrate that QED cascade can develop even in a plane electromagnetic wave. The cascade front propagates as vacuum breakdown waves which is similar an avalanche breakdown at a gas discharge developing via ionization waves. The cascading makes the radiation pressure acceleration inefficient at extremely high intensities. QED cascades may also develop because of Weibel instability in two counterstreaming hot relativistic plasma flows. If the plasma flows are dense, fast, and hot enough, the overall energy of the synchrotron photons can be much higher than the energy of the generated
electromagnetic fields. Furthermore, a sizable number of positrons can be produced due to the pair photoproduction in the generated magnetic field. We propose a rough criterion to judge copious pair production and considerable synchrotron losses. By means of this criterion, we conclude that the incoherent synchrotron emission and the pair production during the Weibel instability can have implications for the collapsar model of gamma-ray bursts.