This is a list of courses that anyone can play with advance reservations, that in my opinion are the most fun I've played. Six are in the UK, where it's easier to access private courses, and five are in the US.
UK
North Berwick - the best of the best in this category!
Elie - close to St Andrews and blessed with a great variety of holes including half a dozen short par 4s - and no par 5's!
Crail Balcomie - seven par 3's, none easy, even closer to St Andrews
Painswick - the poster child for fun on the golf course
Pennard - serious golf but many fun holes, plus a ruined castle!
Ardglass - over to Northern Ireland, crazy clifftop holes and another castle, this one houses the clubhouse
US
Pasatiempo - Mackenzie masterpiece that anyone can play, and the wild greens are fun!
Pacific Dunes - a modern masterpiece meant to be enjoyed, with many ground game options
Aiken Golf Club - what public course golf can be, the surprise hit of the 2009 Dixie Cup
Southern Pines - Donald Ross gem open to the public at a good price, Pinehurst on a budget
Tobacco Road - you have laugh at some of the situations you find yourself in!
Whoops - Longshadow is an obvious oversight. One fun hole after another. The conditions aren't great, but the golf is.
My 10 becomes a dozen, a 20% bonus for Sean!