As with all rodents, Chinchillas' teeth grow throughout their lives, including their front teeth and their molars. From a "survival in the wild" perspective, this constant growth is required to ensure that chinchillas' teeth don't get worn down to nothing from all the chewing their food requires. As pets, we need to make sure our chinchillas have many items to chew on plus their food.
Maintaining Tooth Health with Food
Chinchillas innately know that they need to chew on things continually to keep the growth of their teeth in check. The first step is to ensure that they have teeth-appropriate foods. One example is timothy hay which is quite coarse and reportedly popular with chinchillas and their owners. As a backup, we've heard that high quality meadow hay is also good. Check the hay to make sure it doesn't smell musty or dusty.
Chinchillas should be given as much hay as they want. Unlike with treats, chinchillas won't overeat hay so it's safe to keep a constant supply in their cage. It also makes sense to keep the hay in a rack of some sort attached to the side of the cage to keep it from getting soiled. Hay that has been compressed into cubes is also a good way to keep it from being inadvertently used as bedding.
Other Ways to Maintain Tooth Health
In addition to the right food, you'll want to make sure that chinchillas have enough other things to chew on. Of particular importance is supplying them with safe wood. Cardboard boxes are also good and your chinchilla will enjoy being able to hide in them. For those with a little extra money to lavish on their chinchillas, there are various chew toys available, many made out of wood.
Sources:
eRodent
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Hi Kelly
Take your baby to the vet - one specializing in chins/exotics. My baby Apollo was about 3.5 yrs old when he started to drool. First trip to vet increase timothy hay but didn't do anything. At 4 I took him back and big drool issue - xrays found he had the tooth growth issue (malocclusion) which could be treated by filing down the teeth but bigger problem he had root issues as well that we couldn't do anything about. For 2 years I hand fed and gave him his pain meds and occasionally we had to get his teeth filled down again. It was expensive but we had 2 good years together before it was all too much for him. He left me in June this year but I wouldn't trade those last 2 years for anything.