Rodent
incisors keep growing and growing. Actually, they are fascinating study in the
formation of dental tissue, as well as the characteristic peculiar to continual
eruption.
Yes, the rats tooth keep forming while simultaneously being worn down by use. If the rat did not eat, it would have these impressively long and longer front teeth!
Enamel and dentine, the two hard layers of teeth, are continually being deposited at the apex of the tooth - not unlike hair formation, I guess. The periodontal ligament is unique in that it allows for continual eruption of the tooth. It has to continually attach and detach itself from the tooth, and is only present on one side of the tooth.
If human were to wear their teeth down, eventually the pulp ("the nerve") would be exposed. But in the rat, the continual deposition of dentine within the pulp cavity continues through life so that exposure of the pulp never takes place. This process of secondary dentine formation also occurs in the human tooth, but to a very limited extent.
(sigh... else, where would root canal therapy be?)