One proposed formation channel for stellar mass black holes (BHs) is through hierarchical mergers of smaller BHs. Repeated mergers between comparable mass BHs leave an imprint on the spin of the resulting BH, since the final BH spin is largely determined by the orbital angular momentum of the binary. We find that for stellar mass BHs forming hierarchically the distribution of spin magnitudes is universal, with a peak at a ∼ 0.7 and little support below a ∼ 0.5. We show that the spin

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... is robust against changes to the mass ratio of the merging binaries, the initial spin distribution of the first generation of BHs, and the number of merger generations. While we assume an isotropic distribution of initial spin directions, spins that are preferentially aligned or antialigned do not qualitatively change our results. We also consider a "cluster catastrophe" model for BH formation in which we allow for mergers of arbitrary mass ratios and show that this scenario predicts a unique spin distribution that is similar to the universal distribution derived for major majors. We explore the ability of spin measurements from ground-based gravitational-wave (GW) detectors to constrain hierarchical merger scenarios. We apply a hierarchical Bayesian mixture model to mock GW data and argue that the fraction of BHs that formed through hierarchical mergers will be constrained with O(100) LIGO binary black hole detections, while with O(10) detections we could falsify a model in which all component BHs form hierarchically.