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We present a novel, few-body computational framework designed to shed light on the likelihood of forming intermediate-mass (IM) and supermassive (SM) black holes (BHs) in nuclear star clusters (NSCs) through successive BH mergers, initiated with a single BH seed. Using observationally motivated NSC profiles, we find that the probability of a ${\sim}100 \, M_\odot$ BH to grow beyond ${\sim}1000 \, M_\odot$ through successive mergers ranges from ${\sim}0.1\%$ in low-density, low-mass clusters to nearly $90\%$ in high-mass, high-density clusters. However, in the most massive NSCs, the growth timescale can be very long ($\gtrsim 1\,$Gyr); vice versa, while growth is least likely in less massive NSCs, it is faster there, requiring as little as ${\sim}0.1\,$Gyr. The increased gravitational focusing in systems with lower velocity dispersions is the primary contributor to this behavior. We find that there is a simple "7-strikes-and-you're-in" rule governing the growth of BHs: our results suggest that if the seed survives 7 to 10 successive mergers without being ejected (primarily through gravitational wave recoil kicks), the growing BH will most likely remain in the cluster and will then undergo runaway, continuous growth all the way to the formation of an SMBH (under the simplifying assumption adopted here of a fixed background NSC). Furthermore, we find that rapid mergers enforce a dynamically-mediated "mass gap" between about ${50-300 \, M_\odot}$ in an NSC.