Maximally Symmetric n-Higgs Doublet Models (MS-nHDMs) 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-nHDM obeys an Sp(2n) symmetry, which is softly broken by bilinear scalar masses and explicitly by hypercharge and Yukawa couplings through renormalisation-group effects. The Sp(2n) 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.