Excess of probabilities of elastic processes over inelastic ones is a characteristic feature of the chaotic resonance scattering predicted by the random matrix theory (RMT). Quantitatively, this phenomenon is characterized by the elastic enhancement factor F((β)) that is, essentially, a typical ratio of elastic and inelastic cross sections. Being measured experimentally, this quantity can provide important information on the character of dynamics of the complicated intermediate open system formed on the intermediate stage of various resonance scattering processes. We discuss properties of the enhancement factor in a wide scope from mesoscopoic systems as, for example, heavy nuclei to macroscopic electromagnetic analogous devices imitating two-dimensional quantum billiards. We demonstrate a substantial qualitative distinction between the elastic enhancement factor's peculiarities in these two cases. A complete analytical solution is found for the case of systems without time-reversal symmetry and only a few equivalent open scattering channels.