Why doesn't arm hair keep growing?
It does continue to grow, as does all hair on your body. However, unlike the hair on the top of your head, most of your body hair is finer and, therefore, more fragile. It breaks easier and doesn't become really long.
What happens is that the hair follicles go through cycles where they grow a hair for a time and then stops to rest. During that resting phase, the current hair breaks off. When it starts again, a new hair replaces the old one. The length of time between resting phases control the maximum length an individual hair can reach. The hairs on your arms, legs, and body cycle every few months. The hairs on your head, under your arms, and on your groin cycle every few years. Since each follicle is independent you don't notice that you lose a small hair here or there every once in a while.
Most animals are different, their hair follicles are synchronized so that most of the hair follicles go into a rest phase at the same time. Thus they have periods of shedding, where you and I do not.