Exotic relativistic astrophysical objects–black holes and neutron stars–exhibit a plethora of spectacular and puzzling phenomena. Fascinating observational discoveries made in recent years, including the first images of accreting black holes at the event-horizon scale and the enigmatic Fast Radio Bursts, pose challenges to our understanding of the underlying plasma processes at play. Explaining these phenomena requires understanding how classic nonlinear plasma processes like magnetic reconnection and shocks operate under the extreme conditions expected around black holes and neutron stars, where relativistic, radiative, and QED physical effects become important. Achieving this goal calls for new approaches and ideas that require the combined efforts of researchers from different scientific backgrounds: observational and theoretical high-energy astrophysicists, plasma theorists, computational physicists, and laser-plasma and pulsed-power experimentalists. This conference will bring together a diverse set of participants representing the above disciplines and various career stages. It aims to assess and summarize the existing progress in the area, to facilitate scientific dialog and cross-fertilization between sub-communities, and to identify promising strategic directions for future research, including prospects for novel laboratory studies that could provide key physics insights into the behavior of extreme plasmas.