We establish a sharp estimate for a minimal number of binary digits (bits) needed to represent all bounded total generalized variation functions taking values in a general totally bounded metric space (E, ρ) up to an accuracy of  epsilon > 0 with respect to the L^1-distance. Such an estimate is explicitly computed in terms of doubling and packing dimensions of (E, ρ). The obtained result is applied to provide an upper bound on the metric entropy for a set of entropy admissible weak solutions to scalar conservation laws in one-dimensional space with weakly genuinely nonlinear fluxes. These results have been obtained in collaboration with Khai T. Nguyen (NCSU) and Prerona Dutta (NCSU).