We calculate the amplitude of elastic photon-photon scattering via a single quark loop in the double-logarithmic approximation, presuming all external photons to be off-shell and unpolarized. At the same time we account for the running coupling effects. We consider this process in the forward kinematics at arbitrary relations between t and the external photon virtualities. We obtain explicit expressions for the photon-photon scattering amplitudes in all double-logarithmic kinematic regions. Then we calculate the small-x asymptotics of the obtained amplitudes and compare them with the parent amplitudes, thereby fixing the applicability regions of the asymptotics, i.e., fixing the applicability region for the nonvacuum Reggeons. We find that these Reggeons should be used at x<10-8 only.