Have got and have gotten are different in British and American English. In American English, these two forms have separate meanings, while in British English, have gotten is not used at all. For the details, read the explanations and examples below.
In both British and American English, have got means have when it’s followed by a noun phrase, and have to (or must) when it’s followed by to + a verb. Below are some examples of each. Some of them include contractions with have/has + got, which are common.
HAVE GOT + NOUN = have (more common in British English than American English)
HAVE GOT + VERB = have/has to, or must
Have gotten has three different possible meanings in American English: have obtained, have become, and have entered. Below are some examples of each, including contractions with have/has + gotten, which are common.
HAVE GOTTEN = have obtained
HAVE GOTTEN = have become
HAVE GOTTEN = have entered