Seven Lakes Basin/High Divide Loop (about 18 to 24 miles depending on which route) from trail head at end of Sol Duc River Road. Haven't done it, but it's on my "maybe" list for this summer.
See Green Trails Map 133S (Seven Lakes Basin/Mt. Olympus Climbing), and this archived description from Olympic NPS: http://www.nps.gov/archive/olym/wic/dsoduhidi.htm
Appears easy to combine this looop with other entry/exit points, including valley trails for the Hoh River and the Bogachiel River, if transportation can be arranged.
Shuttle service may be one option -- http://www.windsox.us/
I've read that the loop route is pretty popular. Here's a trip report from 2006: http://www.wta.org/~wta/cgi-bin/wtaweb.pl?7+reports+displayM+2006090312.