[en] The interplay between black holes and fundamental fields has attracted much attention over the years from both physicists and mathematicians. In this paper we study analytically a physical system that is composed of massive scalar fields linearly coupled to a rapidly rotating Kerr black hole. Using simple arguments, we first show that the coupled black hole-scalar field system may possess stationary bound-state resonances (stationary scalar ‘clouds’) in the bounded regime $1<<!--\; <\; -->\mathrm{\mu <!--\; ??\; -->}/m{\mathrm{\Omega <!--\; ??\; -->}}_{\mathrm{H}}<<!--\; <\; -->\sqrt{2}$, where μ and m are respectively the mass and azimuthal harmonic index of the field, and ${\mathrm{\Omega <!--\; ??\; -->}}_{\mathrm{H}}$ is the angular velocity of the black-hole horizon. We then show explicitly that these two bounds on the dimensionless ratio $\mathrm{\mu <!--\; ??\; -->}/m{\mathrm{\Omega <!--\; ??\; -->}}_{\mathrm{H}}$ can be saturated in the asymptotic $m\to <!--\; ???\; -->\mathrm{\infty <!--\; ???\; -->}$ limit. In particular, we derive a remarkably simple analytical formula for the resonance mass spectrum of the stationary bound-state scalar clouds in the regime $M\mathrm{\mu <!--\; ??\; -->}\gg <!--\; ???\; -->1$ of large field masses: ${\mathrm{\mu <!--\; ??\; -->}}_n=\sqrt{2}m{\mathrm{\Omega <!--\; ??\; -->}}_{\mathrm{H}}[1-<!--\; ???\; -->\frac{\mathrm{\pi <!--\; ??\; -->}(\mathcal{R}+n)}{m|\mathrm{l}\mathrm{n}\mathrm{\tau <!--\; ??\; -->}|}]$, where τ is the dimensionless temperature of the rapidly rotating (near extremal) black hole, $\mathcal{R}<<!--\; <\; -->1$ is a constant, and $n=0,1,2,\dots <!--\; ???\; -->$ is the resonance parameter. In addition, it is shown that, contrary to the flat-space intuition, the effective lengths of the scalar field configurations in the curved black-hole spacetime approach a finite asymptotic value in the large mass $M\mathrm{\mu <!--\; ??\; -->}\gg <!--\; ???\; -->1$ limit. In particular, we prove that in the large mass limit, the characteristic length scale of the scalar clouds scales linearly with the black-hole temperature. (paper)