For veggies that are top growing 10 inches is a good minimum, 12 a good average and 18 inches moore than enough. Rooting vegetables need enough room on a common sense basis as you are not going to get 10 pounds of potatoes in an 18 inch pot For long root vegetable - competetive growers for shows etc will often grow them in pieces of vertical 18 inch (or less) drain pipe to get long straight roots.
I agree that virtually any vegetable can be grown in containers, except corn. However, you can grow corn in a raised flower bed. We used to grow cherry tomatoes in hanging baskets, and they actually looked nice and the tomatoes didn't lay in the dirt! If you are going to grow your own veggies, don't forget to grow some of your own herbs, too! We always have a pot of mint handy.
I have grown a number of vegetables in containers - tomatos, lettuce, courgettes, dwarf french beans, raddish, broad beans and leeks. Even strawberries have grown well in special strawberry containers on the patio. The added bonus of growing in containers is that if you have a sunny, sheltered spot in your garden or on your patio you can have fresh, homegrown vegetables well into autumn.
I like growing mixed greens in window boxes, and have great succcess with it. The fruit on tomato plants doesn't seem to get as big when grown in pots vs. directly into the ground, but tomatoes that bear smaller fruit anyway, like cherry tomatoes, seem to do especially well in containers.
I don't think there is any veggie that can't be grown in containers. Potatoes might be problematic, but you can grow them in a garbage bag, just keep piling in dirt as they grow UP. Even melons can be rooted in containers, just spread the vines out along the ground as they grow. I've used 5 gallon buckets and other buckets (from the local bakery which had custard filling in it). Personally I plan on trying aquaponics for my next gardening project.
If you look in gardening catalogs or internet catalogs, find "Midget" varieties. These adapt well to containers. I have grown tomatoes, peppers, eggplants, herbs, cucumbers and cantaloupes successfully in containers. The trick is to find ones suitable, so look for 'bush' or 'midget' varieties. Good luck and happy gardening!