  We explore the spectral properties and behaviour of confining
superexponential potentials. Several prototypes of these highly nonlinear
potentials are analyzed in terms of the eigenvalues and eigenstates of the
underlying stationary Schr\"odinger equation up to several hundreds of excited
states. A generalization of the superexponential self-interacting oscillator
shows a scaling behaviour of the spacing of the eigenvalues which turns into an
alternating behaviour for the power law modified oscillator. Superexponential
potentials with an oscillating power show a very rich spectral structure with
varying amplitudes and wave vectors. In the parity symmetric case doublets of
near degenerate energy eigenvalues emerge in the spectrum. The corresponding
eigenstates are strongly localized in the outer wells of the potential and
occur as even-odd pairs which are interspersed into the spectrum of delocalized
states. We provide an outlook on future perspectives including the possibility
to use these features for applications in e.g. cold atom physics.
