In American English, “gotten” is often “to have received” or “to have acquired.”
I have gotten three packages this week. = I have received three packages this week.
Less likely: I have got three packages this week.
Have you gotten my phone calls? = Have you received my phone calls?
Wrong: Have you got my phone calls?
He has gotten three homes in just five years. = He has acquired or received three homes in just five years.
“Got” is “to have possession of.”
I’ve got the clothes you’re looking for. = They are merely in my possession. I have them.
*I’ve gotten the clothes you’re looking for. = I have actively sought them out and acquired them.
I’ve got your pen. = I have it.
I’ve got three packages. = I have three packages.
I’ve got the answer to your troubles. = I have the answer
As you can see, “gotten” is more active than “got” in the previous two sets.
They are also used to narrow something down to specifics and in time.
I’ve gotten it right three times. -(So far but not necessarily in this moment) What you’ll most often hear
I’ve got it right three times - Less likely
Contrast those two with these:
I’ve gotten it right. - Some time before but not necessarily now (I have had it right before.)
I’ve got it right. - Likely right now (I have it correct in this moment.)
“Got” = demands or necessity:
You have got to stop calling me. = You must stop calling me.
Never: You have gotten to stop calling me.
Have you gotten to travel to Rome? = Have you ever traveled to Rome?
Have you got to travel to Rome? = Must you travel to Rome? or Do you have to travel to Rome?
The past participle of “get into,” especially if you mean it physically, will also take “gotten.”
I have gotten into my home.
Never: I have got into my home.
I have gotten into my classroom.
I have gotten into trouble.
Less likely but not as offensive as Number 2: I have got into trouble.
“Get in,” “get out,” and “get around” all function similarly.
Gotten = “To have become.”
I have gotten less sensitive over the years.
Never: I have got less sensitive over the years.
Gotten = “To have been able to.”
He has gotten to make films for many years. = Has been able to make them for many years.
If I use “got,” the meaning will be confused:
He has got to make films for many years. = Likely to be taken as “He must make films for many years.”
The differences are legitimate – they often change the meaning significantly – so I do appreciate “gotten” for its clarity.