Anchorage has a nice collection of courses. They all have their own personality, which gives you a completely new experience each time you play a new one.
MUST PLAY!
Kincaid Park DGC Anchorage, AK . It's very close to the Anchorage airport, so cut your sightseeing three hours short to play it on your way home. It may be the most perfect disc golf course, but I only allocated two hours, so couldn't hit all the holes. Biggest regret of my trip.
Highly Recommended
Hilltop Ski Area Anchorage, AK. A lot of fun, on a ski hill. The quality of the design hits you in the face.
Fun enough to be worth the time to play if you are there (in order of fun-ness):
Peters Creek DGC Chugiak, AK So many trees you almost get claustrophobia, but the flight lines are all there for you to hit. Or not. A unique experience.
Kenai Eagle DGC Kenai, AK Play through the woods, interesting lines. Not long. Hook up with a local if you can, to get all the secret routes and tees.
Girdwood DGC Girdwood, AK Nothing special, but it's still disc golf.
Mountain DGC @ Alyeska Resort Girdwood, AK Another ski area, but kind of rough around the edges, which detracts from the experience, and only 9 holes. Play it if you stay there.
Russian Jack Springs Anchorage, AK Not the best example of adding disc golf to a golf course, and they charge too much.