Human brain processes sensory information in real-time with extraordinary efficiency compared to the possibilities of current artificial computing systems. It operates as a complex nonlinear system, composed of interacting dynamic units - neurons and synapses - that processes data-streams as time goes by, i.e. through time, using time as an internal self-standing variable. Here we report on a memristor-based compact chaotic circuit included in a computing architecture that can process information through time. We realized a hardware memristive version of the formally simplest chaotic circuit that, thanks to the nonlinearity of the nonvolatile memristor device, evolves with complex dynamics in response to a driving signal. The circuit is used in a single-node reservoir computing scheme to demonstrate nonlinear classification tasks and the processing of data streams through time. These results demonstrate that a simple memristor-based chaotic circuit has the potential to operate as a nonlinear dynamics-based computing system and to process temporal information through time.