In the U.S., what does a one room extension (say 10'x10' or so) typically cost?

It depends. Are you enclosing a sunroom? Will you need plumbing? Electrical? Connections to central heat/air? Are you building off a sliding glass patio door or knocking down an exterior wall to get to the addition? Is your house on a slab or does it have a crawlspace? What materials are being used to finish the outside? The roof? How many windows, and what type/size? What kind of insulation?

Even knowing all the answers to those questions, it will vary depending on where you live, with permits and architects needed to draw up official plans for those permits.

Even a very basic, small addition is going to be around $15,000 (nowhere near a 10x10 like you're looking for), and easily up to $40,000 or even more. If you're relocating your kitchen, be prepared to spend closer to $100,000 after all the fees--and that's before cabinets and appliances.

Don't forget that your property taxes will go up, too.