Luminous spheroids (M V −21.50 ± 0.75 mag) contain partially depleted cores with sizes (R b ) typically 0.02 -0.5 kpc. However, galaxies with R b > 0.5 kpc are rare and poorly understood. Here we perform detailed decompositions of the composite surface brightness profiles, extracted from archival Hubble Space Telescope and ground-based images, of 12 extremely luminous "large-core" galaxies that have R b > 0.5 kpc and M V −23.50 ± 0.10 mag, fitting a core-Sérsic model to the galaxy spheroids. Using 28 "normal-core" (i.e., R b < 0.5 kpc) galaxies and 1 "large-core" (i.e., R b > 0.5 kpc) galaxy from the literature, we constructed a final sample of 41 core-Sérsic galaxies. We find that large-core spheroids (with stellar masses M * 10 12 M ⊙ ) are not simple high-mass extensions of the less luminous normal-core spheroids having M * ∼ 8 × 10 10 − 10 12 M ⊙ . While the two types follow the same strong relations between the spheroid luminosity L, and the spheroid half-light radius R e (R e ∝ L 1.08±0.09 V , for ellipticals plus BCGs), we discover a break in the core-Sérsic σ − L V relation occurring at M V ∼ −23.50 ± 0.10 mag. Furthermore, we find a strong log-linear R b − M BH relation for the 11 galaxies in the sample with directly determined SMBH masses M BH -3/11 galaxies are large-core galaxies-such that R b ∝ M 0.83±0.10 BH . However, for the large-core galaxies the SMBH masses estimated from the M BH − σ and core-Sérsic M BH − L relations are undermassive, by up to a factor of 40, relative to expectations from their large R b values, confirming earlier results. Our findings suggest that large-core galaxies harbour overmassive SMBHs (M BH 10 10 M ⊙ ), considerably (∼ 3.7 − 15.6σ and ∼ 0.6 − 1.7σ) larger than expectations from the spheroid σ and L, respectively. We suggest that the R b − M BH relation can be used to estimate SMBH masses in the most massive galaxies.