Fabrication of unconventional energy storage devices with high stretchability and performance is challenging, but critical to practical operations of fully power-independent stretchable electronics. While supercapacitors represent a promising candidate for unconventional energy-storage devices, existing stretchable supercapacitors are limited by their low stretchability, complicated fabrication process, and high cost. Here, we report a simple and low-cost method to fabricate extremely stretchable and high-performance electrodes for supercapacitors based on new crumpled-graphene papers. Electrolyte-mediated-graphene paper bonded on a compliant substrate can be crumpled into self-organized patterns by harnessing mechanical instabilities in the graphene paper. As the substrate is stretched, the crumpled patterns unfold, maintaining high reliability of the graphene paper under multiple cycles of large deformation. Supercapacitor electrodes based on the crumpled graphene papers exhibit a unique combination of high stretchability (e.g., linear strain ~300%, areal strain ~800%), high electrochemical performance (e.g., specific capacitance ~196 F g⁻¹), and high reliability (e.g., over 1000 stretch/relax cycles). An all-solid-state supercapacitor capable of large deformation is further fabricated to demonstrate practical applications of the crumpled-graphene-paper electrodes. Our method and design open a wide range of opportunities for manufacturing future energy-storage devices with desired deformability together with high performance.