Not much, I suspect. Most of the information the browser is sending is necessary for it to function properly. For kicks, I tried it through an anonymizing proxy server and got only slightly better results: 1 in 18,000 rather than 1 in 110,000. And that was probably mostly due to the proxy shutting off cookies, which I usually have on.

Incidentally, the claim of "several hundred thousand visitors" is dishonest, since if the web browser is totally unique, they reveal that only ~110,000 browsers have been tested so far. I imagine that will go up over the next few days, but still…