Rotating black holes can support quasi-stationary (unstable) bound-state resonances of massive scalar fields in their exterior regions. These spatially regular scalar configurations are characterized by instability timescales which are much longer than the timescale M set by the geometric size (mass) of the central black hole. It is well-known that, in the small-mass limit 伪≡M渭鈮? (here 渭 is the mass of the scalar field), these quasi-stationary scalar resonances are characterized by the familiar hydrogenic oscillation spectrum: , where the integer is the principal quantum number of the bound-state resonance (here the integers l=1,2,3,… and n=0,1,2,… are the spheroidal harmonic index and the resonance parameter of the field mode, respectively). As it depends only on the principal resonance parameter 2c2">, this small -mass (伪鈮?) hydrogenic spectrum is obviously degenerate . In this paper we go beyond the small-mass approximation and analyze the quasi-stationary bound-state resonances of massive scalar fields in rapidly-spinning Kerr black-hole spacetimes in the regime 伪=O(1). In particular, we derive the non-hydrogenic (and, in general, non -degenerate) resonance oscillation spectrum , where c2c9886"> is the generalized principal quantum number of the quasi-stationary resonances. This analytically derived formula for the characteristic oscillation frequencies of the composed black-hole-massive-scalar-field system is shown to agree with direct numerical computations of the quasi-stationary bound-state resonances.