MrOysterhead: expand your options to include the Carters and Wildcats and you can have a great, varied, trip. One of my favorites is as follows:
Day 1: Stony Brook Trail to Imp shelter. Take a side trip to Mt Moriah (4k#1)
Day 2: Imp Shelter to Carter Hut (flame suit on) via Carter Moriah Trail and Carter Dome Trails, bagging Middle Carter (4k#2), South Carter(4k#3), Mt Hight (don;t miss: awesome views), Carter Dome (4k#4).
Day 3: Carter Hut to Hermit Lake Shelter via Wildcat Ridge and Tuckerman trails, bagging Wildcat "A"(4k#5) and "D"(4k#6).
Day 4: Hermit Lake to Crawford Notch via Tuckerman Ravine and Crawford Path bagging Monroe (4k#7), Eisenhower (4k#8) and Pierce (4k#9). If you are feeling particularly strong, you can continue toward Jackson or catch Washington on the way by (both give you 4k#10). I usually skip Washington - it's always easy to talk occosional hikers to head up there for a day trip.
4 days, 9-11 4ks.
Another excellent trip, though with less backpacking:
Day 1: Hike into Great Gulf to the legal campsites near the Wamsutta/Buttress/Great Gulf trail intersection (sorry - don't have my map with me for eact location). Set up basecamp (short day)
Day 2: Head up Buttress Trail to Madison Hut. Bag Madison, then backtrack to Hut. Continue on to bag Adams, Jefferson, Clay and Washington, generally following the Gulfside trails and summit loops. Descend back to campsite via Wamsutta trail back to campsite. This is a long day (I'm guessing about 15 miles), and you need great weather to be safe. Ther are many bail out points along the way (note: this route can be reversed, but I prefer hiking the Northern Presis in a counter clockwise direction - but that's just me.)
Day 3: Get up late, hike out. (short day)
I've left off the side and connector trails on both trips (hey, trip planning as part of the fun!) and mileage, but with good weather, an early start, and strong motivation, all can be done by the average hiker with a full pack. (I've done them, and I;m barely average!)
Enjoy - whjile you are enjoying the Whites, I hope to be doing the same in the ADKs!