We report the discovery of an extremely large (Rb ∼2.77 arcsec ≈ 4.2 kpc) core in the brightest cluster galaxy, IC 1101, of the rich galaxy cluster Abell 2029. Luminous core-Sérsic galaxies contain depleted cores - with sizes (Rb) 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-Sérsic galaxies. We find that the spheroid's V-band absolute magnitude (MV) of -23.8 mag (∼25 per cent of the total galaxy light, i.e. including the stellar halo) is faint for the large Rb, such that the observed core is 1.02 dex ≈ 3.4σs (rms scatter) larger than that estimated from the Rb-MV relation. The suspected scouring process has produced a large stellar mass deficit (Mdef) ∼4.9 × 1011 M⊙, i.e. a luminosity deficit ≈28 per cent of the spheroid's luminosity prior to the depletion. Using IC 1101's black hole mass (MBH) estimated from the MBH-σ, MBH-L and MBH-M* relations, we measure an excessive and unrealistically high number of 'dry' major mergers for IC 1101 (i.e. N ≳ 76) as traced by the large Mdef/MBH ratios of 38-101. The large core, high mass deficit and oversized Mdef/MBH ratio of IC 1101 suggest that the depleted core was scoured by overmassive SMBH binaries with a final coalesced mass MBH ∼ (4-10) × 1010 M⊙, i.e. ∼ (1.7-3.2) × σs larger than the black hole masses estimated using the spheroid's σ, L and M*. The large core might be partly due to oscillatory core passages by a gravitational radiation-recoiled black hole.