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 \alpha \equiv M \mu \ll 1 ( here \mu is the mass of the scalar field ) , these quasi-stationary scalar resonances are characterized by the familiar hydrogenic oscillation spectrum : \omega _ { \text { R } } / \mu = 1 - \alpha ^ { 2 } / 2 { \bar { n } } ^ { 2 } _ { 0 } , where the integer \bar { n } _ { 0 } ( l,n; \alpha \to 0 ) = l + n + 1 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 \bar { n } _ { 0 } , this small -mass ( \alpha \ll 1 ) 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 \alpha = O ( 1 ) . In particular , we derive the non-hydrogenic ( and , in general , non -degenerate ) resonance oscillation spectrum { { \omega _ { \text { R } } } / { \mu } } = \sqrt { 1 - ( \alpha / { \bar { n } } ) ^ { 2 } } , where \bar { n } ( l,n; \alpha ) = \sqrt { ( l + 1 / 2 ) ^ { 2 } -2 m \alpha + 2 \alpha ^ { 2 } } +1 / 2 + n 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 .