The maximum power output and minimum charging time of a lithium-ion battery depend on both ionic and electronic transport. Ionic diffusion within the electrochemically active particles generally represents a fundamental limitation to the rate at which a battery can be charged and discharged. To compensate for the relatively slow solid-state ionic diffusion and to enable high power and rapid charging, the active particles are frequently reduced to nanometre dimensions, to the detriment of volumetric packing density, cost, stability and sustainability. As an alternative to nanoscaling, here we show that two complex niobium tungsten oxides-Nb16W5O55 and Nb18W16O93, which adopt crystallographic shear and bronze-like structures, respectively-can intercalate large quantities of lithium at high rates, even when the sizes of the niobium tungsten oxide particles are of the order of micrometres. Measurements of lithium-ion diffusion coefficients in both structures reveal room-temperature values that are several orders of magnitude higher than those in typical electrode materials such as Li4Ti5O12 and LiMn2O4. Multielectron redox, buffered volume expansion, topologically frustrated niobium/tungsten polyhedral arrangements and rapid solid-state lithium transport lead to extremely high volumetric capacities and rate performance. Unconventional materials and mechanisms that enable lithiation of micrometre-sized particles in minutes have implications for high-power applications, fast-charging devices, all-solid-state energy storage systems, electrode design and material discovery.