Any version where the JL is offset & complimented by lighter/golden age elements (DCAU is a great example, large roster, lots more team dynamics, whole lot of action in a colorful world to tackle on) would be fun to hang out with, and hang with the Titans and the JSA by extension. I can't imagine going through any of the myriad crises would be fun, though.
The Avengers, relatively speaking, don't have as much inter-personal drama (I mean, you could end up in a ham-fisted argument about registration, or figure out your team was built as a distraction while the end of the universe was being handled by a bunch of dudes calling themselves the illuminati.) I imagine if you hung out with a street-version level of the team, for instance Bendis's new Avengers you'd have a decent time, being bankrolled by Stark and playing cards with Spidey, Wolverine, and Luke Cage.