I can recommend Greece, especially if you backpack around and use the train network on mainland Greece, then Island hop, there´s lots of islands.
The culture, food, music, people, architecture, archeoligical sites etc. can take you back to ancient Greece if you have the imagination and interest.
Of course its best to avoid the main tourist beach resorts or your just going to see red drunken people.
There is so much to see in Greece, not just the Acropolis in Athens as many would think, and the people are so mellow and friendly.
I can´t begin to list what there is, what particulary interested me was of course Athens and whats to see there, the Corinth ship canal, the Asclepion on the island of Kos, the vulanic island of Nissyros. Best get a book or go to Lonely planet to get an in depth oversight.
Its a matter of taste and what your looking for of course, hope this helped🙂
It might be a bit too much for just a week, and cost over a grand.
But it´d be well worth it if you can get more time off and loan, mind you if you back pack like lots of people do, Ive met loads of people from various countries there, you can doss on beaches, stay in youth hostels or take a small tent with you and camp at cheap places.🙂