It depends on what type of grass you are talking about, for cool climate grasses like Fescue or Blue grass then 13" should be plenty to suppress the grass, I have no experience with warm climate grasses like Bermuda but from what I understand I think it would probably come up thru the bed. If you wet the bed area down prior to layering cardboard and then wet the cardboard down before filling with soil the cardboard will rot pretty quickly and probably not deter the root penetration.

I used thick layers of newspaper over grass and it worked--cardboard will probably be fine. When you mention adding mulch, do you mean: grass, cardboard, mulch, then topsoil? If so, I'd probably skip the mulch and just add topsoil to the beds.

And, if you are worried about roots (until the cardboard degrades--which would likely happen in time for next year's crop) you could "plant up" the tomatoes and peppers in tall, above-ground pots with the bottoms cut out, to give them more height/root depth. They might need more water this way and the soil will be warmer. I'd probably do both -- half in the bed and half in the pot to see which do better.

How fast the cardboard rots away depends in part on your climate/humidity/temp range. In my climate it rots within a couple of moths so when I make a new bed I use a double layer of it. Further north in cooler climes it likely last longer. Either way i wouldn't hesitate to use it. 13" deep beds (assuming they are filled close to the top rather than just 1/2 way as many seem to do) should do fine even for tomatoes. They will just run laterally if needed.

Dave

I also have a 12 inch deep bed, and grass did come up through all that soil, so I would definitely use cardboard. I don't know what type of grass I have, but I live in northern VA.

Actually, the cardboard works great. It deters grass and weed growth from below and provides a welcome attraction to earthworms that improve your soil. This raised bed has a layer of cardboard under it and it supports artichokes and has supported tomatoes.