As far as the media knows, Johnny Depp did not get braces. The only time he doctored on his teeth was while filming Pirates of the Caribbean: Curse fo the Black Pearl. He asked the dentist to replace most of his pearly white with gold ones. Yet, an executive told him to get the pearly whites all back except a few.
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For the Pirate movies they were colored to look like they were. In real life his teeth are not rotten. But many actors have their teeth capped and/or whitened artifically, Depp doesn't seem to be one of them. He seems to be the type of person who is not obsessed about his looks, which is rare for someone as handsome as Depp.
Yes. Johnny Depp had gold teeth put in for the Pirates of the Caribbean. He decided to keep them iinstead of having them pulled.