  If astrophysical black hole candidates are the Kerr black holes predicted by
General Relativity, the value of their spin parameter must be subject to the
theoretical bound $|a_*| \le 1$. In this work, we consider the possibility that
these objects are either non-Kerr black holes in an alternative theory of
gravity or exotic compact objects in General Relativity. We study the accretion
process when their accretion disk is geometrically thick with a simple version
of the Polish doughnut model. The picture of the accretion process may be
qualitatively different from the one around a Kerr black hole. The inner edge
of the disk may not have the typical cusp on the equatorial plane any more, but
there may be two cusps, respectively above and below the equatorial plane. We
extend previous work on the evolution of the spin parameter and we estimate the
maximum value of $a_*$ for the super-massive black hole candidates in galactic
nuclei. Since measurements of the mean radiative efficiency of AGNs require
$\eta > 0.15$, we infer the "observational" bound $|a_*| \lesssim 1.3$, which
seems to be quite independent of the exact nature of these objects. Such a
bound is only slightly weaker than $|a_*| \lesssim 1.2$ found in previous work
for thin disks.
