  We perform a detailed study of the location of brightest cluster galaxies
(BCGs) on the fundamental plane of black hole (BH) accretion, which is an
empirical correlation between a BH X-ray and radio luminosity and mass
supported by theoretical models of accretion. The sample comprises 72 BCGs out
to $z\sim0.3$ and with reliable nuclear X-ray and radio luminosities. These are
found to correlate as $L_\mathrm{X} \propto L_\mathrm{R}^{0.75 \pm 0.08}$,
favoring an advection-dominated accretion flow as the origin of the X-ray
emission. BCGs are found to be on average offset from the fundamental plane
such that their BH masses seem to be underestimated by the
$M_\mathrm{BH}-M_\mathrm{K}$ relation a factor $\sim$10. The offset is not
explained by jet synchrotron cooling and is independent of emission process or
amount of cluster gas cooling. Those core-dominated BCGs are found to be more
significantly offset than those with weak core radio emission. For BCGs to on
average follow the fundamental plane, a large fraction ($\sim40\%$) should have
BH masses $> 10^{10}$ M$_{\odot}$ and thus host ultramassive BHs. The local
BH-galaxy scaling relations would not hold for these extreme objects. The
possible explanations for their formation, either via a two-phase process (the
BH formed first, the galaxy grows later) or as descendants of high-z seed BHs,
challenge the current paradigm of a synchronized galaxy-BH growth.
