What is a good, relatively inexpensive camcorder that can take videos with depth of field?

You mean a small depth of field.....

The other answer is correct basically for the money youre willing to spend forget it, not possible.

I have a letus adapter which I use on cheap (what I call cheap compared to what a usually use) Sony Z1 on EX1 or EX3 with some cheap slr lenses and it works great you get a tiny depth of field which you can manipulate as much as you want. I bought my letus with lenses and a camera support for £1500 2nd hand on ebay thats around $2000, then I hire the cameras in depending on what my clients want to use. You could buy a canon 550D that's a stills camera that does video and is a cheaper compromise, crap sound though...

the easy answer is none. cheap camcorders use very small CCD pickups, often 1/8 th inch. because the focal length of the lens is proportional to the image size, they use a very short focal length, say 3 mm or less. to get the selective focus/depth of field effect at reasonable distance, say 10 foot from the camera, you need a 100 mm lens. this is no problem for motion pictures because 100 mm is the normal lens for 65 mm movie film. 100 mm also works well with 35 mm movie and still film, usually desired for portraits because the foreshortening effect is flattering to the main subject. The only way to get a 100 mm lens on a camcorder is to use a Letus adaptor or equivalent. The Letus allows you to use the lens from a 35mm SLR or dSLR, meaning that selective focus is practical. The Letus is not cheap, but you can save some by selecting one that has a mount for SLR lenses you may already have. Even the Letus cannot be used with super small CCDs. You may be able to get one for 1/4 inch. A camcorder with 1/4 inch or 1/3 inch CCDs will cost well over $500. The Letus35 mini costs $700. and if you don't have an SLR lens, that will add more than $100 to the total. there are ways of getting a selective focus effect using Cokin filters, but your camcorder will still need to be in the 1/3 inch CCD or larger type. A fog filter will blur detail at a distance, but preserve the appearance of the main subject. A spot diffuser has a clear zone in the center where the subject would be and diffusion to spoil detail elsewhere. you can make your own by buying a clear UV filter for the camcorder and applying vaseline gel around the edges. be sure not to get any gel in the center or on the camera lens.

You can consider Panasonic SDR-SW21 Shock & Waterproof Camcorder (Green)

Triple-tough waterproof, shockproof and dustproof design

Records to SD/SDHC memory cards

10x optical zoom; 25x digital zoom

Easy upload to YouTube

2.7-inch widescreen LCD