In review of the subject, you can think of so many reasons why you can stay in one city for a week. The purpose of the query is to get your recommendation on cities you feel would make a great base camp where my daughter (14) and I can see the local history and do day trips.
I prefer the base camp process versus traveling from city to city every two days or every day. I like to spend the night in the same bed for the week.
My last trip, I spent a week in Rome and a week in Sorrento. I really enjoyed a week in Sorrento and visiting the surrounding area.
I'm open to any city in the world, but I assume my daughter wants to shop also.
The Traveler