I recently picked up and read this crossover made 10 years ago.
I think it's by far the best crossover I've read so far, at least as far as these two big comic companies are concerned. It's about how Krona and Grandmaster bet against the two big teams from their own universes in what is basically a treasure hunt of various important items (i.e. Marvel's Infinity Gauntlet, DC's Lantern Corps Power Battery). The JLA and the Avengers are told of their mission and crossover to each others worlds, and both condemn the others' impact on their universes. For example, Superman is disgusted at how ineffective the Avengers are at preventing disasters like mutant genocide and stopping dictators like Dr Doom, while Captain America regards all the DC heroes as fascists, who demand admiration from an impressionable public.
After the JLA win the tie, Krona is about to raise hell because of his insistence on discovering all of the universe's secrets, until Marvel's and DC's universes are merged into one, as both teams have their memories and histories altered by believing they were close allies.
My favorite is when Grandmaster reveals all of the main heroes' tragic events happening in their lives, including Batman grieving Robin's death and getting injured by Bane, Superman dying after fighting Doomsday, Iron Man's struggle with alcoholism, Aquaman losing his hand etc.
(https://www.batman-online.com/forum/proxy.php?request=http%3A%2F%2Fbenpittdesign.files.wordpress.com%2F2012%2F02%2Fjla_avengers_assemble.png&hash=415374ea23c447131f250eafb3b51ab7309b594d)
Man supes looks pretty boss holding cap's shield :)