  Motivated by complex microstructures in the modelling of shape-memory alloys
and by rigidity and flexibility considerations for the associated differential
inclusions, in this article we study the energy scaling behaviour of a
simplified $m$-well problem without gauge invariances. Considering wells for
which the lamination convex hull consists of one-dimensional line segments of
increasing order of lamination, we prove that for prescribed Dirichlet data the
energy scaling is determined by the \emph{order of lamination of the Dirichlet
data}. This follows by deducing (essentially) matching upper and lower scaling
bounds. For the \emph{upper} bound we argue by providing iterated branching
constructions, and complement this with ansatz-free \emph{lower} bounds. These
are deduced by a careful analysis of the Fourier multipliers of the associated
energies and iterated "bootstrap arguments: based on the ideas from
\cite{RT21}. Relying on these observations, we study models involving laminates
of arbitrary order.
