The emergence of a complex, large-scale organisation of cosmic matter into the Cosmic Web well exemplifies of how complexity can be produced by simple initial conditions and simple physical laws. Connecting the stunning variety of multi-messenger observations to the complex interplay of fundamental physical processes is an open challenge for Big Data in astrophysics. In this contribution, I discuss a few relevant applications of Information Theory to the task of objectively measuring the complexity of modern numerical simulations of the Universe. When applied to cosmological simulations, the metric of complexity makes it possible to measure the total information necessary to model the cosmic web. It also allow us to monitor which physical processes are mostly responsible for the emergence of complex dynamical behaviour across cosmic epochs and environments, and possibly to improve mesh refinement strategies in the future.