You need not worry about infinity in the black-hole case. Known physics ends at the black hole's event horizon, for an external observer. Black holes of a given mass can only have so much angular momentum.

As a rule of thumb, that angular momentum corresponds to that at which the equator of the black hole is moving at the speed of light.

This has important consequences for the mergers of two high-spin black holes that have aligned spins. The merger is actually delayed for a little while, just before merger, as the binary dumps angular momentum through gravitational radiation.