It depends on how you define a week, what you're seeking and how far you want to travel.
There's a newish 40 mile Grafton Loop route in the Mahoosuc Range which will see far less traffic than anything in the Presidential Range:
A word of warning - over the years there has been a fairly consistent problem with vandalism of cars parked at Grafton Notch State Park
For higher milage, this can be combined with the AT along the spine of the Mahoosuc Range for a "lolly-pop"- See
Less ambitious loops can be created around the Wild River Wilderness - this describes a loop along the Carter-Moriah Range and the Wild River. It can be extended by adding the ridges that form the eastern edge of the Wild River basin.
Planning Map for both Wild River and Mahoosuc:
The classic loop centers on the Pemigewasset Wilderness. If you walk over every 4000 foot peak in this area, you'll cover approximately 85 miles and 22000 feet of elevation gain and loss. This includes several out and back spur hikes where you can drop your pack, so its not quite as strenuous as it sounds and is very easily modified. You could easily spend a week here, especially in September when the days are getting short. This wilderness can include several miles of alpine ridgeline that are subject to fairly severe weather - it's a good place to be willing to change plans to fit the weather.
Planning Map for Pemi: http://amcstore.outdoors.org/AB1805000/showdetl.cfm/DID/8/Product_ID/535/CATID/29