We consider the number of Bowen sets necessary to cover a large measure subset of the phase space. This introduces some complexity indicator characterizing different kinds of (weakly) chaotic dynamics. Since in many systems its value is given by a sort of local entropy, this indicator is quite simple to calculate. We give some examples of calculations in nontrivial systems (e.g., interval exchanges and piecewise isometries) and a formula similar to that of Ruelle-Pesin, relating the complexity indicator to some initial condition sensitivity indicators playing the role of positive Lyapunov exponents.