Globally-coupled N-body systems are well-known to possess an intricate dynamics. When N is large, collective effects may drastically lower the effective dimension of the dynamics breaking the conditions on ergodicity necessary for the applicability of statistical mechanics. These problems are here illustrated and discussed through an entropy-complexity analysis of the repulsive Hamiltonian mean-field model. Using a Poincaré's section of the mean-field time series provides a natural sampling time in the entropy-complexity treatment. This approach is shown to single-out the out-of-equilibrium dynamical features and to uncover a transition of the system dynamics from low-energy non-Boltzmann quasi-stationary states to high-energy stochastic-like behavior.