TL;DR as deep as you can.

If you are going to allow the veggies to grow into the native soil below, 12" is probably ok for most things. If you are going to put a weed/root/solid barrier on the bottom, it becomes a container, so (draining issues aside), if you plan to grow tomatoes, squash, asparagus, rhubarb, okra, beans, cucumbers, carrots, onions... you need more than 12". If you can, make then 24".

Agree, 12'' is not enough, unless you have nice soil below your bed. 18'' is min, 24'' is better. I use 8'' wide boards, and 3 boards work fine. The trick is, you do not have to fill whole bed with planting soil, that could be pretty expensive. You can place a 6-8 inches layer of of any dry compostable materials on the bottom(straw, shredded leaves, pine needles, even fine wood chips). They will work as moister reservoir, and will prevent soil to dry out too fast. As you fertilize your bed, excess water with fertilizer will be setting up in the bottom layer also providing the roots nutrients if they overgrow soil level. In one or two season the bottom layer will decompose, lowering the soil level. This will allow you to add more organics like compost to the soil. Good luck!

Um, is there a reason you don't want the roots to go into your soil?

When I start a bed I usually break up as much of our clay soil as I can, add organic manure (horse/chicken/goat/llama, composted),compost (if you have it) water in, water in again until the manure doesn't soak in any more. Build the raised bed with hardware cloth for gophers, add some yummy soil from the expensive yard place (our really inexpensive organic place at the dump is gone!) water/soil/water until full. And actually over fill the first year cause it will all go down in time.Our beds are only 8-10 inches high.

If there is a reason you don't want your roots to go into your soil, you probably want to check out the container growing forum. Nancy

Agree with Nancy - what is wrong with your soil? Odds are no matter how deep you make it some roots will get there anyway. Agree that 18" is minimum and go for 24" if you can.

Dave