Arm strength isn't a limit once you can do 10-20 good chins, unless you mean very long and overhung climbs. If you mean bouldering though, it can be a limit until you can do a 1 arm chin up or 1 arm lock offs (very very rare requirements to do them but such climbs still exist)
Finger strength is almost always a limit until you get to very very high end climbers who can support their entire weight off one finger or pinch their entire weight with two hands (grab the bottom of a ceiling beam and do a chin up).
Aside from the odd bouldering problem massive back strength and a 1 hand pull up are just a personal challenges, really (especially if your finger strength isn't amazing). Finger strength is another story...