Glacier
Glacier National Park is my favorite place to backpack in the lower 48. My friend and I backpacked there for 2 weeks and logged over 200 miles in a complete traverse of the park from Walton (SE corner) to Waterton National Park in Canada. My favorite parts of the trip were the Two Medicine area because is seemed more remote and the scenery was spectacular, and the North Fork/Belly River Area which is also seemingly very remote. I can give you as many specifics as you want as far as which were my favorite campsites, trails, etc. We also climbed a couple peaks when we were there which was an added bonus. We saw a wealth of wildlife on the trail: moose, mule dear, mountain goats, bighorn sheep, and yes grizzley bears. My friend and I were extremely close to a grizzley and could have been killed if the bear had chosen to do so. The mother grizzley was nursing her cub in a bush on the side of the trail. My friend did not realize this and startled the bear at which point she jumped out of the bush to defend her young. The bear looked at us right in our eyes, my friend about 20 feet away, I was about 50 feet away. The bear took 2 steps closer to my friend at which point I thought he was going to be lunch, but lucky for us the mother bear ran uphill off the trail and away from us. This happened to us on the 14th day in the backcountry, and we had been complaining the whole trip about not seeing a bear.
The backcountry in Glacier is very regulated so you need to have you plans solid before applying for permits. Some campsites are not available for reservation until August 15 or later due to snow. Space is very limited so if you do get a permit you won't see many people out there which is nice.