  When two black holes merge in a dense star cluster, they form a new black
hole with a well-defined mass and spin. If that "second-generation" black hole
remains in the cluster, it will continue to participate in dynamical
encounters, form binaries, and potentially merge again. Using a grid of 96
dynamical models of dense star clusters and a cosmological model of cluster
formation, we explore the production of binary black hole mergers where at
least one component of the binary was forged in a previous merger. We create
four hypothetical universes where every black hole born in the collapse of a
massive star has a dimensionless Kerr spin parameter, $\chi_{\rm birth}$, of
0.0, 0.1, 0.2, or 0.5. We show that if all stellar-born black holes are
non-spinning ($\chi_{\rm birth} = 0.0$), then more than 10% of merging binary
black holes from clusters have components formed from previous mergers,
accounting for more than 20% of the mergers from globular clusters detectable
by LIGO/Virgo. Furthermore, nearly 7% of detectable mergers would have a
component with a mass $\gtrsim 55M_{\odot}$, placing them clearly in the mass
"gap" region where black holes cannot form from isolated collapsing stars due
to the pulsational-pair instability mechanism. On the other hand, if black
holes are born spinning, then the contribution from these second-generation
mergers decreases, making up as little as 1% of all detections from globular
clusters when $\chi_{\rm birth} = 0.5$. We make quantitative predictions for
the detected masses, mass ratios, and spin properties of first- and
second-generation mergers from dense star clusters, and show how these
distributions are highly sensitive to the birth spins of black holes.
