Energy storage devices that can deliver high powers have many applications, including hybrid vehicles and renewable energy. Much research has focused on increasing the power output of lithium batteries by reducing lithium-ion diffusion distances, but outputs remain far below those of electrochemical capacitors and below the levels required for many applications. Here, we report an alternative approach based on the redox reactions of functional groups on the surfaces of carbon nanotubes. Layer-by-layer techniques are used to assemble an electrode that consists of additive-free, densely packed and functionalized multiwalled carbon nanotubes. The electrode, which is several micrometres thick, can store lithium up to a reversible gravimetric capacity of similar to 200 mA h g(electrode)(-1) while also delivering 100 kW kg(electrode)(-1) of power and providing lifetimes in excess of thousands of cycles, both of which are comparable to electrochemical capacitor electrodes. A device using the nanotube electrode as the positive electrode and lithium titanium oxide as a negative electrode had a gravimetric energy similar to 5 times higher than conventional electrochemical capacitors and power delivery similar to 10 times higher than conventional lithium-ion batteries.