  The superexponential self-interacting oscillator (SSO) is introduced and
analyzed. Its power law potential is characterized by the dependence of both
the base and the exponent on the dynamical variable of the oscillator. Opposite
to standard oscillators such as the (an-)harmonic oscillator the SSO combines
both scattering and confined periodic motion with an exponentially varying
nonlinearity. The SSO potential exhibits a transition point with a hierarchy of
singularities of logarithmic and power law character leaving their fingerprints
in the agglomeration of its phase space curves. The period of the SSO
consequently undergoes a crossover from decreasing linear to a nonlinearly
increasing behaviour when passing the transition energy. We explore its
dynamics and show that the crossover involves a kick-like behaviour. A
symmetric double well variant of the SSO is briefly discussed.
