The Snow Lakes area is easier to get permits for than the core Enchantments area, and a very worthy visit with a stop at Nada Lake to break up the hike.
The Snow Lakes Trail leaves from the Snow Lakes parking lot on the edge of Icicle Creek Road 5 miles outside of Leavenworth, WA, winding its way up 3,800 feet in elevation over 6.5 miles. Add in another 1.5 miles to reach the far end of the lakes, and another 1.5 miles and 1,400 feet elevation from Upper Snow Lake to Lake Viviane inside the Core Enchantments area if you choose to continue.
Note that permits are required for overnight camping, though not currently for day-hikers. If you have a permit, there's camping at Snow Lakes.
This is a long hike, and not recommended for younger children due to the distance and elevation gain.
Because of the special permit zone in the Enchantments, I haven't yet had the opportunity to camp in any of the various zones. I have, however, been able to do a day-trip up to climb Prusik Peak via Aasgard Pass (amazing, but makes for a looong day, from about 4am to 2am). This time (early June), I went up the Snow Lakes trail sans ropes, stopping for a chilly but beautiful swim at Nada Lake. By the time we arrived at Snow Lakes (only a little snow on the ground), thunder clouds were rolling in and we had to high-tail it back to the car, about 7 miles back the way we came. Less steep than the Aasgard Pass side of the Enchantments, I really enjoyed the hike through dense forest and talus slopes. There was plenty of fresh water along the route, not to mention it was beautifully crystal clear. There were few bugs and the trail was dry. Next time, I really hope to be able to camp up there. ★ ★ ★ ★ ★