Matt,
Great topic! I assume you mean fun other than tallying up the course stats as to yards, par, and course rating?
Sorry, couldn't resist the shot!
I think most golfers have fun (in approximate order) playing a:
Historic course with tournament history
Naturally beautiful course
Museum piece that hasn't changed much
(not as universal)
These three go a long way to determine the golf experience for most, and are a huge advantage. The problem comes when designing a new course with no history obviously, and less than perfect scenery. Then, you need beauty to the degree possilble and:
Course where they don't lose many golf balls
Well maintained course (can get points for fast and firm)
Course with a different look and strong theme - whether quarry, mountain, links, augusta landscaping, Texas Hill Country, etc. Anything but Geneic Country Club (or muni) USA
Course with different type holes you haven't seen much
(Your first Cape, Dell, Redan, whatever. For most golfers, this could also be playing your first course by a certain GCA, as above, one with a different style than your home course(s))
Course with different shots you don't hit much
(not as universal, but a shot where you hit it here to get it there, i.e. use imagination like TePaul says Not everyone will be thrilled with playing out of a road hole bunker, or having a nice (but indifferent) chip roll of the back of the green)
Course with unexpected pleasant surprises, like kick in banks
(Oh Hell, Oh Hell, Oh Hell, it's perfect!)