I've figured out that the packing nut is 3/4". The problem I have now is that it's super deep in there, and the valve stem is really long! The packing nut is set back about 1.5 inches into the wall, and the valve stem sticks out another 2.5 inches out of the wall. So to get at the packing nut, I would need a 4 inch deep socket!
As I mentioned earlier in the thread, I have a shower valve socket set, but it doesn't include a 3/4" size.
Can anyone point me to a 3/4" socket that's deep enough for the job, or maybe suggest some other tool to try?