I believed Mel Bartholomew, too, about the 6-inch depth for Bed #1. What hooey! It sure didn't work for me.
The other beds/boxes are 12 inches deep. They have (had?) newspapers at the bottom, no weed cloth. I've been fighting that...stuff...at my MIL's house since 2007, when the landscapers put it in. Weeds come up through it anyway when it's used between shrubs (full sun, lots of water), and when you pull the weeds up, they don't want to come up, the weed cloth comes up, and you have a real mess.
We built our raised beds/boxes in Spring and Summer 2008. I'm starting to see a few weeds in a couple of them, but nothing like I see in the surrounding area. Seems like being deprived of sunlight (12" underneath the soil) slows them down significantly!
But the neighborhood cats and--yes--squirrels have been running through and digging, and there are many birds as well that like the way I let my veggies go to seed, so weed seeds have no doubt been dropped into the boxes. No problem: the soil is very loose, so pulling the weeds out is pretty easy.
I'd recommend no weed cloth. Larger-rooted veggies will need more room; that's what happened with Box #1 in 2008. I had veggies "on hold" for months because they couldn't get through the not-yet-broken-down newspaper, and there wasn't sufficient soil (only 6") for their roots.
Sunset Zone 17, USDA Zone 9