  The largest observed supermassive black holes (SMBHs) have a mass of M_BH ~
10^{10} M_sun, nearly independent of redshift, from the local (z~0) to the
early (z>6) Universe. We suggest that the growth of SMBHs above a few 10^{10}
M_sun is prevented by small-scale accretion physics, independent of the
properties of their host galaxies or of cosmology. Growing more massive BHs
requires a gas supply rate from galactic scales onto a nuclear region as high
as >10^3 M_sun/yr. At such a high accretion rate, most of the gas converts to
stars at large radii (~10-100 pc), well before reaching the BH. We adopt a
simple model (Thompson et al. 2005) for a star-forming accretion disk, and find
that the accretion rate in the sub-pc nuclear region is reduced to the smaller
value of at most a few M_sun/yr. This prevents SMBHs from growing above
~10^{11} M_sun in the age of the Universe. Furthermore, once a SMBH reaches a
sufficiently high mass, this rate falls below the critical value at which the
accretion flow becomes advection dominated. Once this transition occurs, BH
feeding can be suppressed by strong outflows and jets from hot gas near the BH.
We find that the maximum SMBH mass, given by this transition, is between
M_{BH,max} ~ (1-6) * 10^{10} M_sun, depending primarily on the efficiency of
angular momentum transfer inside the galactic disk, and not on other properties
of the host galaxy.
