Just 3 day hikes for all of yosemite is quite a challenge, but here would be my (>3) suggestions, based largely on the idea that there are three special things about yosemite: sequoias, the valley, and the high country.
If you've never seen giant sequoias, go to the mariposa grove with time to hike well into the upper grove which is the nicest part-- I think if you really explore everwhere, it's close to 10 miles in a figure 8 returning to the parking lot. During high season, they run these dopey open air buses on roads through the park, but I think if you go later in the day, you can at least have the return trip without these. It's not at all big mountain scenery, but it's is amazing. Be aware that it's about a 60-90 min drive from the valley to get to the mariposa grove trail head.
A nice high country loop trip is to leave from Tolumne meadows up Lyell canyon to the Ireland Creek trail to vogelsang high sierra camp, then back via the raferty creek trail for a triangular loop. the first leg of this trip is a nearly flat walk up a meadowy glaciated canyon. The middle leg (Ireland creek) steady uphill through the forest, and then the upper point (evelyn lake and voglesang high sierra camp) a beautiful subalpine meadow and lake area. The trip down raferty creek is more subalpine meadow giving way to granite benches and lodgepole pine forest. It's been a long time since I've done this loop, and I think it might be a bit more than your 15 mile limit, but it's nice. If you come in June and there's still snow on the ground, you might get lucky and have the high sierra camp yet to open. If not, it's a bit of a jolt to have hiked several hours and then run in to the alpine zoo up there.
Another beautiful loop in this area that is much shorter but involves a bit of (easy) cross country is to go up the Budd lake non-trail (this branches off from the cathedral lake trail that leaves from Tolumne meadows. it's not marked or official, but it's in the guidebooks and is plain as day on the gound). Unlike the official trails in the area, this is more of a singletrack foot path, rather than constructed horse highway, and it stays relatively close to budd creek. It goes ~3.5 miles to Budd lake. To continue to Cathedral lake, backtrack a few hundred yards, and look for a little side path marked with a duck going towards the obvious saddle with cathedral peak. even if you don't see the path, just head for the saddle-- it's easy, open, pretty travelling. The saddle itself is broad and open, then you go down open forests and slabby granite to Cathedral lake, which is beautifully nestled under cathedral peak. This is ~3+ miles via the john muir trail back to your starting point (this is the trail that you left earlier to head to budd lake). Cathedral is a great place to watch the alpenglow on the peak and reflections on the lake, and the trail back is so big and obvious that it's easy to follow back with a headlight if you were to stay and watch the sunset. I'm not sure what the total milage on this trip would be, but <10, and with the exception of the last bit through the forest at the bottom of the cathedral trail, most of it is in really scenic terrain.
People criticize the Valley for the crowds, which can be immense, but it's a truely unique, special place-- the big trees, high mountains, etc... can all be found other places, but not the valley. the mist trail, up past vernal and nevada falls, is often crowded, and it's paved for the first mile or two because it's so high use, but all these caveats aside, it's spectacular. One way that you can make this into a figure 8 rather than an out and back is to go up the mist trail, then return via the panaroma tral, and before crossing the footbridge below vernal fall, look to your left below the bathrooms to hit the horse trail back, which takes you back along the other side of the river, through a swampy area, past a big water tank, and then back to the happy isles nature center where you started. This isn't marked on most trail maps, but is on the topos.
--another option would be to continue up to glacier point. From here, you can ride a shuttle bus back to the valley (or walk the 4-mile trail back down).
--a third option, which I would do if it were me, is to go up the mist trail to the top of nevada fall, and spend any extra time and energy scrambling around up Liberty cap, the giant dome that towers above you there. See:
http://www.summitpost.org/view_object.p ... _id=151618" onclick="window.open(this.href);return false;
--a fourth variation on the mist trail is to go up the abandoned trail up to Sierra Point. note that this leaves from the paved "official" trail to vernal fall, NOT the horse trail I mentioned above. See here for directions:
http://www.summitpost.org/route/160150/ ... point.html" onclick="window.open(this.href);return false;
People often talk about climbing Half Dome as
the hike to do if you're in yosemite. I strongly disagree-- the beginning (the mist trail) is nice, and the end, going up half dome, is spectacular, but in between is a long boring slog. But if you really want to be on top, then I guess it's worth it. As previus posts noted, clouds rest is a better view (all the same big views, but including Half Dome!), nicer hike, and not the same crowds. One option, also taking advantage of the various shuttle busses, would be to go from the sunrise trailhead to clouds rest and then half dome, finishing via the mist trail into the valley. I'm not sure how long this would be-- probably closer to 20 than 15 miles-- and the ideal way to start, say at 4am, probably isn't feasable with the suttle busses, but it might be worth looking into. I also think it's hard to really appreciate how crowded half dome can be in the middle of the day-- see:
http://www.summitpost.org/image/330948/ ... rowds.html" onclick="window.open(this.href);return false;
the best way to avoid this, and be up on top when it's nice light in the morning, is to either hike through the night or camp en route.
In terms of crowds: The high country trails (lyell canyon-raferty loop, budd lake, cathedral, clouds rest) will have other people on them, but hardly "crowds". The cross-country section between budd and cathedral would likely be just you and perhaps the distant yells of the climbers on Cathedral Peak. The mist trail is mellow early in the morning, and not so bad up past the top of vernal fall, but in the middle of the day and afternoon is truely crowded from Happy Isles to Vernal Fall. Scrambling on Liberty cap or Sierra POint would likely be just you. In the Mariposa grove, you're unlikely to ever be alone, but it's not exactly crowded-- but I've never been there when the busses are running.
Note that among the other hassles of a crowded park is that it can be difficult to get campground reservations. While **I** would obviusly never condone illicit behavior, a less constrained would-be car camper willing to walk a few hundred yards with their stuff could always cook dinner at a picnic area and then crash at the Tolumne Meadows or Yos. Valley backpackers camp with the excuse that you just arrived and are going to get a wilderness permit at the ranger station when they open the next morning. Although after dinner you'd be unlikely to see any ranger there anyway.