I can think of several great trails to spend 3 weeks on. It depends on what shape you are in, and what your goals are.
Certainly the Long Trail is a good choice provided that you are in reasonable (good) shape, and want to do an end to end. I should caution you that north of Middlebury Gap, the Long Trail becomes the Long and Hard Trail.
You might consider the AT in Maine. If you are in hiking shape, it takes about 3 full weeks to go from Gorham to Katahdin.
For that matter, you could do 3 weeks just about anywhere on the trail. Each section of trail is unique and has it’s good points.
You might consider the Northville-Lake Placid Trail in the Adirondacks. That’s shorter and does not go over the high peaks. Probably about a 10 day trip by itself.
Or spend some time bagging the 46 High Peaks of the Adirondacks. That is strenous hiking, and many peaks are trail-less. You will probably want a car to go from trail head to trail head.
Or work on the 4000 footers in New Hampshire. Again, you will probably want a car to go from trail head to trail head.
So many trails, So little time.
Peaks