  In this work, we study a prototypical, experimentally accessible scenario
that enables the systematic generation of so-called high-order rogue waves in
atomic Bose-Einstein condensates. These waveforms lead to significantly and
controllably more extreme focusing events than the famous Peregrine soliton. In
one spatial dimension, we showcase conclusive numerical evidence that our
scheme generates the focusing behavior associated with the first four rogue
waves from the relevant hierarchy. We then extend considerations to anisotropic
two-dimensional and even three-dimensional settings, establishing that the
scheme can generate second order rogue waves despite the well-known limitation
of finite-time blow up of focusing nonlinear Schr\"odinger equations.
