I have been looking at some of my old classic Traveller campaign notes over the last few days. One of the quirks of classic Traveller was that characters did not improve. They started out with a lot of prior experience and abilities and were played with what they had. If you are used to games where leveling up is the goal, classic Traveller seems strange. As we were used to OD&D, I remember thinking the game wouldn't be any fun without regular character improvement before we started playing. That turned out not to be true. Traveller was a lot of fun to play, you just focused on different goals, goals like paying the monthly payment on your ship or tracking down that patron who had left you to hang out to dry when the Imperium came sniffing around.
Many people seem to find that they enjoy playing D&D most at (or near) a particular level of power. They don't really enjoy working up to that level and once they get too far beyond that sweet spot, they find their interest in the character waning. If your fellow players and DM are willing, why not borrow the "no advancement" idea from classic Traveller and just create characters at your group's sweet spot and play them at that level. They can have goals other than advancement, perhaps moving into or up the noble hierarchy, stopping some great evil, or even just becoming rich and famous from their adventures.
Fixed level characters might also present the players with challenges that they do not normally face. For example, if a powerful monster needs to be stopped and it is too powerful to defeat in combat, players will have to think of some other way to defeat it because they are as strong as they will get, they can't just wait until their characters gain a few more levels where can easily defeat it.
I know this idea needs some fleshing out, but I wanted to get it down in writing before it completely slipped my mind. It should work with any edition of D&D (or any other advancement oriented game system). Comments are welcome.