We report the discovery of an extremely large (R-b similar to 2.77 arcsec approximate to 4.2 kpc) core in the brightest cluster galaxy, IC 1101, of the rich galaxy cluster Abell 2029. Luminous core-Sersic galaxies contain depleted cores - with sizes (R-b) typically 20-500 pc -that are thought to be formed by coalescing black hole binaries. We fit a (double nucleus) + (spheroid) + (intermediate-scale component) + (stellar halo) model to the Hubble Space Telescope surface brightness profile of IC 1101, finding the largest core size measured in any galaxy to date. This core is an order of magnitude larger than those typically measured for core-Sersic galaxies. We find that the spheroid's V-band absolute magnitude (M-V) of -23.8 mag (similar to 25 per cent of the total galaxy light, i.e. including the stellar halo) is faint for the large R-b, such that the observed core is 1.02 dex approximate to 3.4 sigma s (rms scatter) larger than that estimated from the R-b-M-V relation. The suspected scouring process has produced a large stellar mass deficit (M-def) similar to 4.9 x 10(11)M(circle dot), i.e. a luminosity deficit approximate to 28 per cent of the spheroid's luminosity prior to the depletion. Using IC 1101' s black hole mass (M-BH) estimated from the MBH-sigma, MBH-L and MBH-M* relations, we measure an excessive and unrealistically high number of `dry' major mergers for IC 1101 (i.e. N greater than or similar to 76) as traced by the large Mdef/MBH ratios of 38-101. The large core, high mass deficit and oversized M-def/M-BH ratio of IC 1101 suggest that the depleted core was scoured by overmassive SMBH binaries with a final coalesced mass M-BH similar to(4-10) x 10(10)M(circle dot), i.e. similar to (1.7-3.2) x sigma(s) larger than the black hole masses estimated using the spheroid's sigma, L and M-*. The large core might be partly due to oscillatory core passages by a gravitational radiation-recoiled black hole.