I grew some giant chard that was biennial. It's a great green, and yielded tons of seeds its second season. Salad rocket is another good self seeder that's loose in my garden. Dandelion and wild lettuce of course. Yellow dock is another one that is ever present around here.
Celery is one plant I have just had go to seed and want to see and taste more of.
At my place:
Arugla (rocket), both annual Eruca saliva which self-seeded in the hoop house, and perennial Diplotaxis tenuifolia which transported to my new property on an empty clay pot.
Lettuces got into everything the year I wanted to see what the flowers looked like. That took about 2 years to grub out (pre-permie tidy gardening)
Chard has come back in the hoop house too. I guess things kind of got out of hand last summer but I have some actual experience on the power of neglect.
Like most of you I love volunteers or self-seeded plants. I almost always get volunteers from red Hopi amaranth (Amaranthus cruentus), vine spinach or basella (Basella rubra) , quail grass (Celosia argentea), rice beans (vigna umbellata)spider wisp (Cleome gyandra), purslane (Portulaca oleracae), orach (Atriplex hortensis), shiso (Perilla fructans), and hyacinth beans (Lablab purpureus), as well as some of the ones others have mentioned. The edge of the compost piles is always a rich zone for free plants. One of the great things about self-seeders is they select for success in your conditions and they decide when is a good time to sprout, saving you some guesswork.
a wee bit from the empire
Ernie and Erica Wisner's Rocket Mass Heater Everything Combo