The black hole GRS 1915+105 was observed to be spinning at near the theoretical maximum spin rate.
Assuming you had two such black holes spinning clockwise, and also orbiting each other clockwise, what would happen if they merged? How could the new black hole avoid breaking the rotational speed limit? Where would the extra angular momentum go?
The simple answer is that no, black holes cannot merge into a new black hole with a spin greater than the maximum allowed spin. The reason being that such an act would cause the black hole to show us a "naked singularity". Black holes have an event horizon which shields their singularity from observation by the outside universe. As they spin faster, that event horizon shrinks. At some point, it completely disappears, causing the singularity to become "naked" and observable to the universe. Some believe that naked singularities cannot possibly form, mainly because if they could, causality and physics would break down. This concept is known as the cosmic censorship hypothesis.
I found a statement concerning this matter in Numerical Relativity: Solving Einstein's Equations on the Computer, which I believe is relevant (emphasis mine).
Black hole spins that are aligned with the orbital angular momentum increase the binary's total angular momentum. If this total angular momentum exceeds the maximum angular momentum of a Kerr black hole, then the binary cannot merge until a sufficient amount of angular momentum has been radiated away. Quite generally, we expect binaries with black hole spins aligned with the orbital angular momentum to merge more slowly than binaries with spins that are anti-aligned. This effect, sometimes referred to as "orbital hang-up", has been explored with numerical simulations.
One such numerical simulation looked at aligned and anti-aligned mergers and found that the aligned black holes took much longer to merge and radiated away more energy, in the form of gravitational waves, before they merged.
There is another way to burn off some of that spin angular momentum too. When spinning black holes merge, they experience a "kick" in their linear momentum. In other words, they suddenly speed up in their motion through space. This kick is a result of converting some of the two distinct black holes' orbital and spin momentums into the merged black hole's linear momentum. Quoting the same book as above concerning these kicks:
Most of the initial calculations focused on black hole spins that are aligned or anti-aligned with the orbital angular momentum. The resulting kicks are several hundred km/s in magnitude, easily exceeding the maximum kick of approximately 175 km/s found for non-spinning black holes.
In summation, two spinning black holes cannot merge into a black hole which is spinning faster than the maximal rotation. Gravitational waves and the linear momentum "kick" play an important part in helping binary black holes to lose spin energy, such that the merged black hole doesn't exceed a maximal rotation.
I think the really interesting question that should be asked is, how do two black holes know, before merging, that their merger cannot happen as it would violate the cosmic censorship hypothesis? How do they know they need to radiate away extra energy before they can merge? What is the mechanism that prevents them from merging? Only time can tell for now.
I believe the orbital angular momentum would need to radiate away in gravitational waves as the merger proceeded. I don't know if any of the spin angular momentum would also radiate away, or if you'd still end up with a maximally rotating black hole. That's probably an oversimplification, one should be skeptical of anything but a full calculation. Note, for example, that we would not want to overlook the kinetic energy of the orbit when we calculate the mass-energy of the system, so if some of that mass-energy ends up in the final black hole mass, it might not seem maximally rotating any more. So although I'm not giving a calculation, I'm saying two things: 1) orbital decay will produce gravitational waves that might carry away angular momentum, and 2) orbital energy counts as mass-energy for the ultimate black hole, so it might not be maximally rotating even if you forget gravitational radiation.
