The growth of large-scale cosmic structure is a beautiful exemplification of how complexity can emerge in our Universe, starting from simple initial conditions and simple physical laws. Using {enzo} cosmological numerical simulations, I applied tools from Information Theory (namely, statistical complexity) to quantify the amount of complexity in the simulated cosmic volume, as a function of cosmic epoch and environment. This analysis can quantify how much difficult to predict, at least in a statistical sense, is the evolution of the thermal, kinetic and magnetic energy of the dominant component of ordinary matter in the Universe (the intragalactic medium plasma). The most complex environment in the simulated cosmic web is generally found to be the periphery of large-scale structures (e.g. galaxy clusters and filaments), where the complexity is on average $sim 10-10^2$ times larger than in more rarefied regions, even if the latter dominate the volume-integrated complexity of the simulated Universe. If the energy evolution of gas in the cosmic web is measured on a $approx 100 $ $rm kpc/h$ resolution and over a $approx 200$ $rm Myr$ timescale, its total complexity is the range of $sim 10^{16}-10^{17} rm ~bits$, with little dependence on the assumed gas physics, cosmology or cosmic variance.
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