  We investigate where brightest cluster galaxies (BCGs) sit on the fundamental
plane of black hole (BH) activity, an established relation between the X-ray
luminosity, the radio luminosity and the mass of a BH. Our sample mostly
consists of BCGs that lie at the centres of massive, strong cooling flow
clusters, therefore requiring extreme mechanical feedback from their central
active galactic nucleus (AGN) to offset cooling of the intracluster plasma
(L_mech>10^44-45 erg/s). Based on the BH masses derived from the M_BH-sigma and
M_BH-M_K correlations, we find that all of our objects are offset from the
plane such that they appear to be less massive than predicted from their X-ray
and radio luminosities (to more than a 99 per cent confidence level). For these
objects to be consistent with the fundamental plane, the M_BH-sigma and
M_BH-M_K correlations therefore seem to underestimate the BH masses of BCGs, on
average by a factor of 10. Our results suggest that the standard relationships
between BH mass and host galaxy properties no longer hold for these extreme
galaxies. Furthermore, our results imply that if these BHs follow the
fundamental plane, then many of those that lie in massive, strong cool core
clusters must be ultramassive with M_BH>10^10M_sun. This rivals the largest BH
masses known and has important ramifications for our understanding of the
formation and evolution of BHs.
