Let's say I start adding electrons to a black hole. As more are
added the black hole starts to develop a charge. The charge increases
as the electrons are added. At some point, will the electrons stop
being drawn into the black hole?
[[Mod. note --
As soon as the BH has any negative charge, there will be an electrostatic
repulsion between the BH and any nearby electrons. There's a limit to
how much charge you can put on a black hole and still have it be a black
hole: charged black holes always have |Q| < M in dimensionless units, i.e.,
a body with |Q| > M in dimensionless units isn't a black hole -- it's either
a solid object or it has a naked singularity at r=0.
I think (but am not quite sure) that |q|/m for electrons is large enough
that the electrostatic repulsion would be > the gravitational attraction
long before the BH charg reached |Q| = M in dimensionless units. I.e.,
I think (but am not quite sure) that the answer to your question is "yes".
for more on the spacetime geometry of a charged black hole.
See Misner, Thorne, and Wheeler exercise 31.8 for more on the author's
question, and for why |Q| > M isn't a black hole any more.