The memristor1,2 is a promising building block for next-generation non-volatile memory3, artificial neural networks4-7 and bio-inspired computing systems8,9. Organizing small memristors into high-density crossbar arrays is critical to meet the ever-growing demands in high-capacity and low-energy consumption, but this is challenging because of difficulties in making highly ordered conductive nanoelectrodes. Carbon nanotubes, graphene nanoribbons and dopant nanowires have potential as electrodes for discrete nanodevices10-14, but unfortunately these are difficult to pack into ordered arrays. Transfer printing, on the other hand, is effective in generating dense electrode arrays15 but has yet to prove suitable for making fully random accessible crossbars. All the aforementioned electrodes have dramatically increased resistance at the nanoscale16-18, imposing a significant barrier to their adoption in operational circuits. Here we demonstrate memristor crossbar arrays with a 2-nm feature size and a single-layer density up to 4.5 terabits per square inch, comparable to the information density achieved using three-dimensional stacking in state-of-the-art 64-layer and multilevel 3D-NAND flash memory19. Memristors in the arrays switch with tens of nanoamperes electric current with nonlinear behaviour. The densely packed crossbar arrays of individually accessible, extremely small functional memristors provide a power-efficient solution for information storage and processing.