Here we demonstrate a full-cell battery design that bridges the energy density and rate capability between that of supercapacitors or pseudocapacitors with that of traditional lithium-ion batteries. This is accomplished by pairing an anode that enables ultrafast ion co-intercalation, an open framework cathode that allows rapid ion diffusion, and linear ether based electrolyte that sustains cell-level stability and high rate performance. We show this platform to be suitable for both sodium and potassium batteries using graphite as the co-intercalation anode, and Prussian blue as the open framework cathode. Our devices exhibit active material energy densities >100 W h kg-1 with power density >1000 W kg-1 with cycling durability approaching ∼80% energy density retention over 2000 cycles. This work brings together state-of-the-art concepts for fast-charging batteries into a full-cell configuration.