An electronic device may have a fiber composite friction hinge with first and second members that rotate relative to each other about a hinge axis. The fiber composite hinge may have first and second sets of interdigitated elongated fingers. Holes in the elongated fingers may receive a shaft of a fastener. The fastener may be tightened to squeeze the elongated fingers together along the hinge axis. The fiber composite hinge may be formed from a fiber composite material such as carbon fiber composite material. Fibers in the composite material may be embedded in binder such as polymer binder. The fibers may wrap around the holes and fold back on themselves in the fingers. Actuators may be provided to adjust the positions of portions of the hinge members. The electronic device may have a flexible layer such as a flexible display that overlaps the hinge structure.