  The maximal ADM mass for (mini-)boson stars (BSs) -- gravitating solitons of
Einstein's gravity minimally coupled to a free, complex, mass $\mu$,
Klein-Gordon field -- is $M_{\rm ADM}^{\rm max}\sim M_{Pl}^2/\mu$. Adding
quartic self-interactions to the scalar field theory, described by the
Lagrangian $\mathcal{L}_I=\lambda |\Psi|^4$, the maximal ADM mass becomes
$M_{\rm ADM}^{\rm max}\sim \sqrt{\lambda}M_{Pl}^3/\mu^2$. Thus, for mini-BSs,
astrophysically interesting masses require ultra-light scalar fields, whereas
self-interacting BSs can reach such values for bosonic particles with Standard
Model range masses. We investigate how these same self-interactions affect Kerr
black holes with scalar hair (KBHsSH) [1], which can be regarded as (spinning)
BSs in stationary equilibrium with a central horizon. Remarkably, whereas the
ADM mass scales in the same way as for BSs, the \textit{horizon mass} $M_H$
does not increases with the coupling $\lambda$, and, for fixed $\mu$, it is
maximized at the "Hod point", corresponding to the extremal Kerr black hole
obtained in the vanishing hair limit. This mass is always $M_{\rm H}^{\rm max
}\sim M_{\rm Pl}^2/\mu$. Thus, introducing these self-interactions, the black
hole spacetimes may become considerably "hairier" but the trapped regions
cannot become "heavier". We present evidence this observation also holds in a
model with $\mathcal{L}_I= \beta|\Psi|^6-\lambda|\Psi|^4$; if it extends to
\textit{general} scalar field models, KBHsSH with astrophysically interesting
horizon masses \textit{require} ultra-light scalar fields. Their existence,
therefore, would be a smoking gun for such (beyond the Standard Model)
particles.
