What does it mean when a dog constantly licking and will not stop?

It means that the dog loves you. it's their way of showing love to you.

ANSWER 2: If a dog is constantly licking itself in a particular spot, that indicates there is pain that the dog is trying to relieve. Frequent licking happens routinely among dogs with Arthritis. The dog, however, should be seen by a veterinarian who can determine the actual cause of the animal's discomfort.