Cool core clusters of galaxies require strong feedback from their central AGN to offset cooling. We present a study of strong cool core, highly-luminous (most with Lx ≥ 1045 erg s−1), clusters of galaxies in which the mean central AGN jet power 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. 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 with, on average, Lkin/Lnuc exceeding 200. Explanations of this puzzle carry significant implications for the origin and operation of jets, as well as on establishing the importance of kinetic feedback for the evolution of galaxies and their surrounding medium.