I have raised beds that are just 6" deep. They sit on top of an area of old concrete, which is cracked and breaking up, so the drainage works fine.
But in 6" depth, last year I grew strawberries, tomatoes, courgettes, onions, shallots, garlic, mini carrots, celery, french beans, mange tout. They all did extremely well and gave me a very good crop. That depth will place some restrictions on you, because you won't be able to grow any root crops that need more depth (parsnips, carrots), and you obviously won't be able to grow anything perennial other than a small plant like strawberries. You also have to be a bit creative about any plant supports, because you can't just stick a cane in the ground and expect it to stand up on its own in that depth. But other than that, I've been growing just about anything in them.
One option might be to do a few separate raised beds rather than one big one. You could do some deeper, for the deep root crops, and some at 6" for the other crops. That would save you some money, if you're paying for top soil, for example, but would allow you to grow just about anything.