  Maximally Symmetric $n$-Higgs Doublet Models (MS-$n$HDMs)define very economic
settings that enable sharp Higgs-sector predictions beyond the Standard Model
(SM) potentially testable at high-energy colliders. The scalar potential of a
MS-$n$HDM obeys an $\mathrm{Sp(2}n)$ symmetry, which is softly broken by
bilinear scalar masses and explicitly by hypercharge and Yukawa couplings
through renormalisation-group effects. The $\mathrm{Sp(2}n)$ also ensures
natural SM alignment and allows for quartic coupling unification up to the
Planck scale. As typical examples, we consider maximally symmetric realisations
of the Type-II 2HDM and the Type-V 3HDM. We show how in terms of a few input
parameters, definite predictions for the entire scalar mass spectrum of the
MS-2HDM and MS-3HDM are obtained, including the SM-like Higgs-boson couplings
to the gauge bosons and fermions.
