What happens if I were to add angular momentum to a maximally spinning black hole? I am aware that a "naked singularity" is not possible. However, what would happen if I added angular momentum to a black hole near the maximal angular momentum limit? Does it asymptote?
How would you add angular momentum? 

The only way to do that is to drop something with angular momentum (relative to the centre of mass of the BH and the thing, if you want to get picky) into the BH. Could be a spinning object, or an object fired in at a oblique angle, or circularly polarised light or even suitably polarised gravity waves.  

Whatever you do, you increase both the mass and the angular momentum of the BH. Now the limit on the spin of a black hole is expressed in terms of a relation between these two quantities and the Schwarzschild radius of the black hole (which is determined by the mass) and it turns out that whatever you add, and however you add it (if you're firing something in obliquely, it has to actually hit the BH so it can't be too ar from the centre of mass) you cannot increase the angular momentum enough relative to the increase in mass to break that limit.

The equations you have to look at are on wikipedia see especially this section. One way of expressing the limit is given there as 
$GM^2 &lt; Jc$  where $J$ is the angular momentum. Suppose you have a black hole just under this limit and add an object of mass $m$ angular momentum just under $Gm^2/c$ velocoty just under $c$ and on a trajectory that just grazes the Schwarzchild radius of the hole at $\frac{2GM}{c^2}$. The angular moment of the result is just under $$GM^2/c + 2GMmc/c^2 + Gm^2/c$$ where the terms are respectively the original angular moment of the hole, the angular moment from the grazing impact and the angular momentum of the added body. This works out to be (just under) $$G(M+m)^2/c$$, the limit for the (now-enlarged) black hole.