Is 'gotten' the past tense form of 'got' or is there even a word gotten?
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Got is already a past tense. It is the past of 'get'. I got some bread at the shop, tells of something that happened in the past.
Got is both a past tense and a past participle. In its past participle form, I have got a blue car, got signifies possession - I am in possession of a blue car
Thus far all of the above is the same both in Britain and America.
Where got refers to the process of acquisition, in the perfect and pluperfect Americans say I have just gotten a copy from the library, whilst we would use got. This also applies in reported speech She told me she had gotten married versus British She told me she had got married.
When it comes to direct reporting of acquisition, where Americans will say: Since we last met I have gotten anew car, British will often avoid got altogether and say something like: Since we last met I have acquired/bought/obtained a new car.