Justice League of America and The Avengers are where DC and Marvel Comics keep their most expensive toys: DC has Superman, Batman, Wonder Woman and many others, while Marvel has Ant-Man, Hawkguy and ... uh, Jann of the Jungle, I think? There might be a couple more. Obviously, the very idea of putting both teams together is enough to make fanboys and accountants alike cream their pants, which is why the companies decided to make it happen in 1983. However, editorial disputes and the sheer logistics of getting so many trademarked logos on the same page caused massive delays, and the crossover only came out in 2003, in a vastly different form.
All the other superheroes were dead that month, so Superman had to fill in for them.
But wouldn't it be awesome if the writers had said "f**k it, let's do a crossover without telling our bosses"? Yes, it would, and we know this because it happened a bunch of times in the '70s -- The Avengers and the JLA would "coincidentally" fight obvious copycats of the other team on the same month, or a story that started in an issue of DC's Aquaman would sneakily continue in Marvel's Sub-Mariner. This was possible due to two reasons: 1) the publishers didn't always bother to read the books back then (especially the "guy who talks to fish" ones), and 2) while the companies hated each other, the creators were all pals and would sometimes party together. In fact, the best example of a secret crossover involves the writers doing exactly that inside the comics.