I have carrots and parsnips that came up from seeds from plants that went to seed last year. I also have several potato plants that came up from small potatoes that were left in the garden when we dug them last fall. Also, the birds scatter the sunflower seeds all over the garden so we have them all over the garden. And then there's the borage...we grew it because the bees like the flowers. That stuff reseeds its self everywhere...
I pull most of the borage out because it gets big if I don't and would take over the garden. I let some of it grow every year to attract the bees to the garden.