If you are looking for a campsite for your next backpacking adventure, I've got three "must do" recommendations. My criteria include (1) the campsite must be a site specifically permitted by either the Forest Service or National Parks, (2) must have a privy, (3) be located along either a designated Forest Service or National Parks trail and (4) have a "spectacular" setting.
3. Lake Ingalls - The hike to Lake Ingalls is a fun adventure, 4.5 miles one way with 2,500 feet of elevation gain and a real "scramble" the final 1/4 mile up to the lake. Fortunately, you won't need to carry your full pack up to the lake since there is no camping here. The designated camping areas are located along a side trail right after you summit Ingalls pass and before you reach the lake. They are spread out over a fairly large area providing lots of privacy and making it not feel crowded. Here's the view from my campsite.

The views of Mt. Stuart and North Cascades are pretty amazing. As a destination, the lake is impressive.

The only disadvantage is that camping is located about 1/2 to 3/4 mile from the lake, making it somewhat inconvenient to go back and forth. On the other hand, you will often find mountain goats in the area - a real treat and good reason for keeping campers some distance away.
2. Snow Lake (Mt. Rainier NP) - There are several lakes named Snow Lake in the area, including the one in the Alpine Lakes Wilderness trail out of Snoqualmie Pass and my favorite camp site of the same name in Mt. Rainier National Park. There are only two campsites here, so you have plenty of privacy. These campsites are also located at the north end of the lake and away from the main trail crowds. At jsut 2.5 miles roundtrip with only 610 ft of elevation gain, the hike is perfect for young families. What impressed me most about this site is the amazing view looking south across the lake from one of the campsites. My mouth dropped when I first saw it.

The biggest advantage to camping here, I believe, is all of the area around the lake available for exploring - including Bench Lake and all the area around the Snow Lake. There is not much of a view from Snow Lake of Mt. Rainier. To get a glimpse, you have to work your way around to the southeast part of the lake (upper left in the above picture).

The real bonus is Bench Lake, only 0.7 miles from the trailhead. This is one of the best views of sunset you will find in the Park.

1. Shriner Peak (Mount Rainier NP) - This is my all time favorite backpacking camp for Judy and I. Just 8.5 miles roundtrip, you climb 3,434 ft. to an old fire lookout. Given the climb, this trail is one of the least traveled in the park but the views are spectacular. Here's the view from our tent at sunrise.

In addition to Mt. Rainier not shone is this pic, you can see both Mt. Adams and Mt. St. Helens to the south. We were blessed with great weather and views staying just above the clouds. Here's the view at sunset.

If you are into night photography, this is a must do location. The Milky Way as I opened our tent flap:

Another panorama above the clouds.

On this night, we even got a little of the northern lights.

One other advantage to camping here is that it won't be crowded. There are only two campsites. However, you won't find any water up here either and by late July and August, the closest stream is nearly back at the trailhead, so you will have to carry a heavy back. It's well worth it though.
We are blessed in the northwest.