One of those little indications of the difference between American and British English is the verb to get. In American (and Canadian) English its past-simple form is got (I got a new car yesterday), and the past participle is gotten (I’ve gotten better at grammar). In British English, got is used for both forms (I’ve got better at grammar). One exception is using have got to refer to possession, (I’ve got three kids). As this structure is the present perfect simple, strictly it should be I’ve gotten three kids, but that would sound like you’ve bought some kids, and might buy some more. Plus, the real meaning of I’ve got… in this context is a present simple meaning, despite using the present perfect simple, so demanding that one stick to the normal structure of the present perfect simple would be a bit silly.
Gotten has crept into British English a little, probably due to increased exposure to American English in the media and online. In Ireland, I think it’s long been more common than got, though we’ve always been more mid-Atlantic in our English. One phrase which seems to predate the use of gotten in British English though, is ill-gotten gains. Why should this phrase, still in use in British English, use gotten and not got? One simple explanation is that it sounds better. Ill-gotten gains has a nice rhythm of alternating stressed and unstressed syllables. You could say ill-got gains, and stress ill and gains, but it wouldn’t sound as smooth.
The real reason though, is that gotten is the original past participle of to get in British English. It began to fall out of fashion by the 18th century, but was picked up in the fledgling United States. This may have been due to the present of the similarly-patterned verb to forget (forgot, forgotten). I think it may also due to the American accent’s use of a glottal stop to represent the letter t. Ending a word in a glottal stop (e.g. got) can be a bit abrupt, but in gotten it flows nicely into the second syllable. So when you hear someone complain about Americanisms, remind them that it was probably British first!