As gravitational wave detectors improve, observations of black hole (BH) mergers will provide the joint distribution of their masses and spins. This will be a critical benchmark to validate formation scenarios. Merging binary BHs formed through isolated binary evolution require both components to be stripped of their hydrogen envelopes before core-collapse. The rotation rates of such stripped stars are constrained by their surface critical rotation, restricting their angular momentum content at core-collapse. We use stripped star models at low metallicities (, and ) to determine the spins of BHs produced by critically rotating stellar progenitors. To study how such progenitors can arise, we consider their formation through chemically homogeneous evolution (CHE). We use a semianalytical model to study the final spins of CHE binaries, and compare our results against available detailed population synthesis models. We find that above BH masses of , the dimensionless spin of critically rotating stripped stars ()) is below unity. This results in an exclusion region at high chirp masses and effective spins that cannot be populated by binary evolution. CHE can produce binaries where both BHs hit this limit, producing a pile-up at the boundary of the excluded region. Highly spinning BHs arise from very low-metallicity CHE systems with short delay times, which merge at higher redshifts. On the other hand, the contribution of CHE to merging binary BHs in the third observing run of the LVK collaboration is expected to be dominated by systems with low spins () which merge near redshift zero. Owing to its higher projected sensitivity and runtime, the fourth observing run of the LVK collaboration can potentially place constraints on the high spin population and the existence of a limit set by critical rotation.