Luminous spheroids (MV ≲ −21.50 ± 0.75 mag) contain partially depleted cores with sizes (Rb) typically 0.02–0.5 kpc. However, galaxies with Rb > 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 Rb > 0.5 kpc and MV ≲ −23.50 ± 0.10 mag, fitting a core-Sérsic model to the galaxy spheroids. Using 28 "normal-core" (i.e., Rb < 0.5 kpc) galaxies and one "large-core" (i.e., Rb > 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* ≳ 1012M☉) are not simple high-mass extensions of the less luminous normal-core spheroids having M* ∼ 8 × 1010–1012M☉. While the two types follow the same strong relations between the spheroid luminosity LV and Rb (), and the spheroid half-light radius Re (, for ellipticals plus Brightest Cluster Galaxies), we discover a break in the core-Sérsic σ–LV relation occurring at MV ∼ −23.50 ± 0.10 mag. Furthermore, we find a strong log-linear Rb–MBH relation for the 11 galaxies in the sample with directly determined supermassive black hole (SMBH) masses MBH—3/11 galaxies are large-core galaxies—such that . However, for the large-core galaxies the SMBH masses estimated from the MBH–σ and core-Sérsic MBH–L relations are undermassive, by up to a factor of 40, relative to expectations from their large Rb values, confirming earlier results. Our findings suggest that large-core galaxies harbor overmassive SMBHs (MBH ≳ 1010M☉), considerably (∼3.7–15.6σ and ∼0.6–1.7σ) larger than expectations from the spheroid σ and L, respectively. We suggest that the Rb–MBH relation can be used to estimate SMBH masses in the most massive galaxies.