I read about a guy on a forum not too long ago who set a goal of bringing his arm measurement from 17" up to 18". I'm pretty sure that I know this guy and if memory serves me correctly, he has only a pull-up bar to work with. Now, I'd comment on on the forum but I got banned (for reasons that are unclear and unexplained to me). So, I thought it would make a great topic here. Plus, I have experience throwing on size to my own arms (a little over an inch, if memory serves me).
All pull-ups and chin-ups work the biceps to varying degrees. What you want to look for when using pulls and chins which bring your arms closer together, near the centerline of your body. This arrangement forces your arms to work harder and doesn't allow the back to help out as much.
for close-grip pulls and chins.
A common mistake when training with pull-ups is dropping out the grip work in the exercise. For most people, the weakest part of their pull&chin game is their grip. So, they use straps or they use a "meat hook" grip when doing the exercise. I don't recommend doing this unless you have to. The trouble is that the biceps tie into some of the muscles in the grip via fascia. The biceps also serve to supinate the forearm. So, you lose bicep work when you seek to mitigate the grip work. A good way to work around this is to mix your grip. That is, have one palm facing towards you and the other palm facing away. I used this grip quite a bit after hyperextending my right thumb.
If you haven't guessed what my favorite pull-up for bicep work is, then you haven't been paying attention. I really like my version of the towel pull-up. If you're new to the Bodyweight Files, here's my video:
There are other approaches. You could use some of the progressions for one-arm chin-up work. Such options includes spreading out the arms to shoulder-width apart and bringing your chin over to one hand and then the other. Another, harder way is to do pulls&chins with one hand holding onto a towel, below the hand holding onto the bar. As they get easier, you simply lower the hand on the towel.
The beauty of pulls&chins is that it'll probably take a long time before you get to the point where it ceases to build muscle and becomes a strength endurance exercise. My experience concurs with Clarence Bass on the topic of reps: you can build up as much muscle size by doing exercises that require 20 reps as you can by doing exercises that require only 5 reps to reach fatigue. The only difference is that the latter builds more max strength than the former. Both build about the roughly the same amount of muscle size though. So, you can stick with most all of the exercises that I've just described. Variety reigns supreme. don't stick to just one of these flavors. Experiment, change it up, and enjoy bigger arms!