Got vs. Gotten

David Crystal, on the gotten/got distinction in The Cambridge Encyclopedia of the English Language (p.311):
“Gotten is probably the most distinctive of all the American English/British English grammatical differences, but British people who try to use it often get it wrong. It is not simply an alternative for have got. Gotten is used in such contexts as
They’ve gotten a new boat. (= obtain)
They’ve gotten interested. (= become)
He’s gotten off the chair. (= moved)
But it is not used in the sense of possession (= have).
AmE does not allow
*I’ve gotten the answer or
*I’ve gotten plenty.
but uses I’ve got as in informal BrE. The availability of gotten does however mean that AmE can make such distinctions as the following:
They’ve got to leave (they must leave) vs
They’ve gotten to leave (they’ve managed to leave).”
(thanks English Grammar FAQ!)