  The properties of selfinteracting boson stars with different scalar
potentials going beyond the commonly used $\phi^4$ ansatz are studied. The
scalar potential is extended to different values of the exponent $n$ of the
form $V \propto \phi^n$. Two stability mechanism for boson stars are
introduced, the first being a mass term and the second one a vacuum term. We
present analytic scale-invariant expressions for these two classes of equations
of state. The resulting properties of the boson star configurations differ
considerably from previous calculations. We find three different categories of
mass-radius relation: the first category resembles the mass-radius curve of
selfbound stars, the second one those of neutron stars and the third one is the
well known constant radius case from the standard $\phi^4$ potential. We
demonstrate that the maximal compactness can reach extremely high values going
to the limit of causality $C_\text{max} = 0.354$ asymptotically for
$n\to\infty$. The maximal compactnesses exceed previously calculated values of
$C_\text{max}=0.16$ for the standard $\phi^4$-theory and $C_\text{max}=0.21$
for vector-like interactions and is in line with previous results for solitonic
boson stars. Hence, boson stars even described by a simple modified scalar
potential in the form of $V \propto \phi^n$ can be ultra compact black hole
mimickers where the photon ring is located outside the radius of the star.
