I guess it depends on what era you're talking about. The early JLA didn't seem to have an arch enemy at all. They had a few regulars, but no Dr Doom or Magneto. I think in more recent years, DC has pushed Darkseid as the League's arch enemy. But even he hasn't had the exposure that most arch enemies get.
Personally, I don't think there is an arch enemy for the League. As dangerous as Darkseid is, he just doesn't face the League often enough to be an arch enemy. And Luthor is more Superman's foe. Who else is there? Amazo, Vandal Savage, Grundy.....the Royal Flush Gang? It's actually kind of pathetic how poor the League's rogues gallery really is, given how legendary the League's core dozen members are in the industry. It says something about the quality of writing coming out of DC, that their premier team has to poach villains from other titles to fill out their enemies list.
Their Archenemy is Starro. Almost Every run of JLA has Starro as a villain within 3-6 arcs. He is a threat to the entire league when he shows up.
Now the argument against is that Starro is not iconic enough nor has a personality to Arch.
But Every potential foe has some issue with why they are not a perfect fit as the JLA's arch.