Fava beans are the only vegetable I had success with for the first couple of years.
Cardoons do well with little care. They were my second success.
I think the third was mountain papaya. Thrives with little care but took a few years to bear.
Chard grows but I don't like it. I finally killed it on purpose.
Perennial buckwheat is easy and makes a very well-flavored spinach substitute.
I finally have some perennial kale going - ot maybe it's normal kale that I'm harvesting as a perennial. I can't keep purple tree collards going. Daubenton's kale might do OK if it ever got any water.
Chilacayote thrives but is next to useless.
I got a little chicory and a few peas, a coupla string beans, some celery.
Pellitory-of-the-wall is pretty intervention-free, but quite boring. I got nettles established but someone requested I remove them.
I have established dandelions and eat them regularly, so at this point they are among the easiest.
Runner beans are pretty easy.
Rhubarb does OK.
I get a few potatoes from some I planted years ago.
I get a small crop of yacon pretty dependably.
I have failed radishes, squash (summer and winter both), onions, jerusalem artichokes (topinambour), turnips, tomatoes, arugula, lettuce, chayote, amaranth, yams, sweet potatoes, purslane and probably several others.
Mulberry leaves and grape leaves are easy, but grape leaves are fairly special-purpose.
Sochan is succeeding, slowly. I imagine it will pick up in a year or two.
Violet leaves are easy, but not great. I tend to forget to use them. Maybe that's because they're in the front yard and I go in back to harvest dinner.
Asparagus is pretty easy to keep alive, but hard to get to be prolific.
Artichokes ditto.
Several people mentioned aloe. How do you use aloe as a vegetable?