  Square-root topology describes models whose topological properties can be
revealed upon squaring the Hamiltonian, which produces their respective parent
topological insulators. This concept has recently been generalized to
$2^n$-root topology, characterizing models where $n$ squaring operations must
be applied to the Hamiltonian in order to arrive at the topological source of
the model. In this paper, we analyze the Hofstadter regime of
quasi-one-dimensional (quasi-1D) and two-dimensional (2D) $2^n$-root models,
the latter of which has the square lattice (SL) (known for the Hofstadter
Butterfly) as the source model. We show that upon increasing the root-degree of
the model, there appear multiple magnetic flux insensitive flat bands, and
analytically determine corresponding eigenstates. These can be recast as
compact localized states (CLSs) occupying a finite region of the lattice. For a
finite flux, these CLSs correspond to different harmonics contained within the
same Aharonov-Bohm (AB) cage. Furthermore, as the root-degree increases, a
kaleidoscope of butterflies is seen to appear in the Hofstadter diagram, with
each butterfly constituting a topologically equivalent replica of the original
one of the SL. As such, the index $n$, which uniquely identifies the
root-degree of the model, can be seen as an additional fractal dimension of the
$2^n$-root model present in its Hofstadter diagram. We discuss how these
dynamics could be realized in experiments with ultracold atoms, and measured by
Bragg spectroscopy or through observing dynamics of initially localized atoms
in a quantum gas microscope.
