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Most stellar evolution models predict that black holes (BHs) should not exist above approximately $50-70$ M$_\odot$, the lower limit of the pair-instability mass gap. However, recent LIGO/Virgo detections indicate the existence of BHs with masses at and above this threshold. We suggest that massive BHs, including intermediate mass black holes (IMBHs), can form in galactic nuclei through collisions between stellar-mass black holes and the surrounding main-sequence stars. Considering dynamical processes such as collisions, mass segregation, and relaxation, we find that this channel can be quite efficient, forming IMBHs as massive as $10^4$ M$_\odot$. This upper limit assumes that (1) the BHs accrete a substantial fraction of the stellar mass captured during each collision and (2) that the rate at which new stars are introduced into the region near the SMBH is high enough to offset depletion by stellar disruptions and star-star collisions. We discuss deviations from these key assumptions in the text. Our results suggest that BHs in the pair-instability mass gap and IMBHs may be ubiquitous in galactic centers. This formation channel has implications for observations. Collisions between stars and BHs can produce electromagnetic signatures, for example, from x-ray binaries and tidal disruption events. Additionally, formed through this channel, both black holes in the mass gap and IMBHs can merge with the supermassive black hole at the center of a galactic nucleus through gravitational waves. These gravitational wave events are extreme and intermediate mass ratio inspirals (EMRIs and IMRIs, respectively).