Plasma-filled magnetospheres can extract energy from spinning black holes and power a variety of observed astrophysical phenomena. These magnetospheres are described by the highly nonlinear equations of force-free electrodynamics, or FFE. Typically these equations can only be solved numerically. In this talk I will explain how to analytically obtain several infinite families of exact solutions of the full nonlinear FFE equations very near the horizon of a maximally spinning black hole, where the energy extraction takes place.