Enabling fast ion diffusion in thick electrodes (100–200 µm, ~ 10 mg cm−2) is critical for their practical application in state-of-the-art supercapacitors (SCs). We developed a three-dimensional (3D) boron, nitrogen, and phosphorus ternary-doped holey graphene hydrogel (BNP-HGH) film to achieve an optimized porous structure with a high electrical conductivity, large ion accessible surface area, efficient electron and ion transport pathways, as well as high ion adsorption capacity. The binder-free BNP-HGH electrode can deliver a specific capacitance of 350 F g−1 and a volumetric capacity of 234 F cm−3, which are the best performance reported so far for graphene-based SCs using an organic electrolyte. Fully packaged SCs using the BNP-HGH electrodes with a commercial level graphene mass loading (150 µm, ~ 10 mg cm−2) can deliver ultrahigh stack gravimetric and volumetric energy densities of 38.5 Wh kg−1 and 57.4 Wh L−1, respectively, which are comparable to those of lead-acid batteries (35–40 Wh kg−1 and 50–90 Wh L−1) while maintaining an ultrahigh power density of 83 kW kg−1 (~ 55 kW L−1) as well as a long cycle life (81.3% capacitance retention over 50,000 cycles). The high energy and power densities bridge the gap between traditional SCs and batteries, and should be very useful in practical applications.