We present a study of strong cool core, highly luminous (most with LX≥ 1045 erg s−1) clusters of galaxies in which the mean jet power of the central active galactic nucleus must be very high yet no central point X-ray source is detected. Using the unique spatial resolution of Chandra, a sample of 13 clusters is analysed, including A1835, A2204 and one of the most massive cool core clusters, RXCJ1504.1−0248. All of the central galaxies host a radio source, indicating an active nucleus, and no obvious X-ray point source. For all clusters in the sample, the nucleus has an X-ray bolometric luminosity below 2 per cent of that of the entire cluster. Most have a nucleus 2–10 keV X-ray luminosity less than about 1042 erg s−1. We investigate how these clusters can have such strong X-ray luminosities, short radiative cooling times of the inner intracluster gas requiring strong energy feedback to counterbalance that cooling and yet have such radiatively inefficient cores. If the central black holes have masses ∼109 M⊙ then the power exceeds 1 per cent of the Eddington luminosity, and they are expected to be radiatively efficient. Only if they are ultramassive (MBH > 1010 M⊙), would their behaviour resemble that of lower mass accreting black holes. Our study focuses on deriving the nucleus X-ray properties of the clusters as defined in the above question, while briefly addressing possible solutions.