Place des Vosges is the oldest square in Paris.
It's also one of the prettiest. I could imagine that finding a place to live on the square would be akin to scoring a place on Gramercy Park (and getting a key to said park).
Victor Hugo's house (and now a museum devoted to him) is here. The buildings have covered galleries over the sidewalk with lots of tiny shops and cafes. I remember the first time I happened upon this neat little square and falling in love with it. It's still one of my favorite places to visit in the city.
The park is filled with boxy manicured linden trees and neat little lawns. There are signs saying you couldn't walk on the lawns, so we followed what we saw Parisians doing and just parked our derrieres on the lawn.