Recently I just read Avengers/JLA, and I flipped through a copy of the newish New 52 version of JLA, mostly because of the Jim Lee art. Granted, I've always been much more of a Marvel guy than a DC guy, but DC has done some things right. In particular, I think DC has had some great animation over the last few years. I recently watched season 1 of Young Justice, I watched Crisis on Two Earths, Death of Superman, the Wonder Woman animation, Batman's Red Hood movie, and Batman/Superman Apocalypse--which is really all about Supergirl, despite the odd name. I also not long ago saw the Superman and Captain Marvel (Shazam) take on Black Adam show, and one of those shows (I can't remember which) even had a Green Arrow story attached to it.
So, in some ways, the JLA concept is growing on me. But curiously, I find the idea of ersatz JLAs much more interesting than the real one. The Squadron Supreme, by Marvel--in both their original and their rebooted "MAX Imprint" versions are very interesting takes on the JLA story. Both of them posit a group of superheroes that's basically an exact copy of the JLA in terms of having characters with corresponding powers and roughly corresponding backgrounds (Hyperion=Superman, Power Princess=Wonder Woman, Nighthawk=Batman, Doctor Spectrum=Green Lantern, Whizzer/Blur=Flash, etc.) but these guys are more interesting as characters. In addition to having more intriguing flaws, they are also much more serious about what they do. Especially the MAX Squadron, from the so-called Supremeverse--these guys do not hold back in the interest of the gentlemanly expectations of comic book fights.
Another fascinating ersatz Justice League was put together by DC themselves; the Crime Syndicate of America on pre-Crisis Earth-3 (later rebooted as Earth-2.) Here, the heroes and villains are reversed; Lex Luthor, the Joker, etc. are all heroes, while counterparts to the JLA (Ultraman=Superman, Owlman=Batman, Superwoman=Wonder Woman, Johnny Quick=Flash, Power Ring=Green Lantern, etc.) are all supervillains.
I'd love to see these alt.JLAs get more play somehow. It seems unlikely that they will in a significant way, but the Supremeverse line was at least a major step forward. I also just reread Ultimate Power, which is the Supremeverse coming into conflict with the Ultimate Universe prior to Ultimatum. It's a 9-issue miniseries published in trade paperback. The first three issues are written by Brian Michael Bendis (co-creator of the Ultimate universe, althlough curiously not the part that is most featured here. Mark Millar would be that person.) The second three are written by J. Michael Straczynski, the main writer and creator of the Supremeverse version of the Sqaudron Supreme, and the final three are written by Jeph Loeb who's... uh... the guy who blew up the Ultimate universe, mostly. Yeah, his issues were the weakest, which kinda sucked, because they were the last three.
What would my ideal crossovers of JLA and ersatz Justice Leagues look like? Probably JLA vs the original Squadron Supreme as a retro-Silver Age style book, and then Supreme Power Squadron vs. the Crime Syndicate as a very dark comic. And then, the Squadron Supreme (either version) needs to make some kind of apperance in the new Avengers Cartoon. Maybe, as in what happened with the Supremevese and Ultimate Universe crossover, Power Princess can come be a part of the Avengers for a time.
None of these things will actually happen, of course.
But a guy can dream.
And the final of those actually could happen, in theory. It all depends on how long the Avengers cartoon gets to run, and how comfortable Marvel Animation is with putting out characters who are very obviously pastiches of a Direct Competitor's own characters.
Not that they haven't already put those characters out there, but this is a whole new medium, and I can imagine the lawyers get cagey.