I'd second Wyoming's Wind River Range--my idea of the most beautiful place on earth. Having grown up in Wyoming, I'm of course prejudiced. I would have gone back there to live after retirement except for the winters!

The Highline/Fremont trail is about 80 miles from Green River Lakes to Big Sandy Opening. (It's also part of the Continental Divide Trail.) Most of it is at or above timberline, so very Alpine. However, there are so many wonderful side trips to take that you could spend 2-3 months there (which is about all you can spend there anyway without being heavily involved with winter.) Those are all in the book that BZH linked to! There's also a High Route, although it requires some climbing/glacier skills (up to Class 3 in a few spots). A few areas have a lot of people (Titcomb Basin, Cirque of the Towers, and close to the main trailheads), but mostly you'll find the Winds delightfully uncrowded, especially if you get even a short way off the main trails.

If you have the $$$, you can have a horse packer bring supplies in (quite possible on the west side of the mountains). Otherwise, you can hike out to Elkhart Park and hitch into Pinedale for resupply, which a lot of folks also do. There is a shuttle service through the Great Outdoors Shop in Pinedale.

My plan, which I was prevented from doing by injuries and family issues the past 3 years, was to take three separate 8-9 day trips from three separate trailheads, keeping most of the food resupply in my car and taking a night in town between each to indulge in fresh food, a beer and a shower. I might still do it, if I can!