Many games are addicting because you just want to reach that next level. You see it again and again in many MMORPG's, text based MMO's, and other games. Stack Exchange already has a leveling system, where you get new privileges at certain 'levels'. Why not make that more visible and therefore more addictive, displaying a progress bar at the top of the page showing a percentage of the points that you need to get the next level, and displaying a level next to a user's reputation on his profile. You could even call the levels 'ranks', and have ranks based on the subject of the site. For example, ranks on stack overflow could start of with "hello world", "conditional", "loop", "exception catcher",...
This partially already exists. You can see your progress towards different privilege levels on the Privileges page.
I would be worried about straying a little into the territory of "exploiting people" here. It's one thing to make SO useful and rewarding via gamification to encourage participation; it's another to focus on making it "more addictive". This idea doesn't seem to add any value, only addictiveness.
I believe I read something relevant on Jeff's blog recently, but I can't find it. I'm sure the powers that be have thought about this line as well.