Do black holes have a limit of mass?

The rules of classical general relativity say that when you add mass to a black hole, you get a larger black hole. If you add angular momentum to a black hole at a greater rate than that at which you add mass, it would theoretically be possible to get a Kerr black hole with $a \gt M$, which would convert the black hole to a naked singularity, but the rules of black hole thermodyanamics say that a black hole with $a = M$ has zero temeperature, so creating a naked singularity in this way is believed to be impossible.

There is the tantalising possibility that adding more matter to a black hole can destroy it, by making its horizon disappear. Indeed a Kerr-Newman black hole, i.e. one that spins and that has an electric charge, must satisfy the relation $m^2 \ge a^2 + e^2$ where $m$ is the mass, $e$ the charge, and $a=J/m$ is the angular momentum per unit of mass (geometrised units G = c = 1 being used throughout). So as @Jerry-Shirmer stated, if the added matter contributes more spin or more charge than mass, then the black hole could be destroyed. There is a long history of research into proving that this cannot actually happen when properly modelling the motion of the matter falling into the black hole. The oldest result is, I think, that of Wald [1] about falling test particles (that is to say, loosely, bits of matter small enough that they do not affect the geometry of spacetime). The question of whether more realistic in-falling matter could destroy a blackhole has been discussed for many years. The latest publication I know of is another paper by Wald(!) [2] and the answer is no! You will find all the bibliography you want in that one. The only assumption is that the matter satisfies the so-called null energy condition, which is valid for all matter we have observed.

[1] Wald, Robert (Feb. 1974). “Gedanken experiments to destroy a black hole”. In: Annals of Physics 82.2, pp. 548–556.

[2] Wald, Robert M. (Aug. 2018). “Kerr–Newman black holes cannot be over-charged or over-spun”. In: Inter- national Journal of Modern Physics D 27.11, p. 1843003.