A new measurement of fusion reactions down to 10 nb has been performed for the system 64 Ni + 64 Ni. Coupled-channels analyses have been carried out for the systems 60 Ni + 89 Ya nd 64 Ni + 64 Ni. They demonstrate that coupled-channels calculations are unable to reproduce the unexpected steep falloff of the cross sections at extreme sub-barrier energies. Heavy-ion fusion excitation functions are also analyzed in terms of the S factor, as this offers a pragmatic way to study fusion behavior in the energy regime of interest. It is shown that the steep falloff in cross section observed in several heavy-ion systems translates into a maximum of the S factor. The energies where the maximum occurs can be parameterized with a simple empirical formula. The parameterization, which is derived here for rather stiff heavy-ion systems, provides also an upper limit for reactions involving softer nuclei. The asymptotic behavior of reaction cross sections at very low energies is a critical issue for calculating reaction rates of astrophysical interest. Recently, it was pointed out that fusion cross sections for several heavy-ion systems show an unexpected behavior at very low energies, 1) with a much steeper falloff than predicted by coupled-channels calculations, 2) or from Wong’s formula. 3) In the present work, we first present a new result for the excitation function in an open-shell system, 64 Ni + 64 Ni down to the 10 nb level, we then analyse in a coupled-channels description the fusion of 60 Ni + 89 Ya nd 64 Ni + 64 Ni, i.e. the systems which have been measured to extreme sub-barrier energies. Fusion reactions at extreme sub-barrier energies are then studied in a representation in terms of the S factor. This parametrization, which in the past has been successfully applied to fusion reactions with lighter ions at low energies, proves to be an alternative and pragmatic way of characterizing and exploring the behavior of heavy-ion fusion cross sections in the energy domain of interest. To our surprise, a maximum appears systematically in the evolution of S with decreasing energy for all systems that exhibit the steep falloff in cross section described above. This maximum occurs at rather high excitation energies in the compound nucleus.