Not sure how many domains are actually visited directly like that - I'm guessing it would be impossible to know exactly - but what would be a rough estimate?
Practically zero. Because:
Update: I'll add a bit based on the good comments and answers given by others in this thread.
I believe all webmasters should do the right thing, and normalize their URLs with HTTP redirects to their preferred site address, with or without the "www." prefix as they see fit.
With prefix normalization in place, the whole argument about typing domain names directly versus searching is moot. In all cases, "www." or not; search or direct input; the user will arrive at the right place in the first try.
To use or to omit "www." based on personal taste, fashion whims, or perceptions of what is common in your target market is completely okay. Just help the users who do the opposite with a redirect.
What I disagree with is the notion that we should all remove "www." because of the external ''rational'' motives of efficiency or usability. ASCII characters are not an endangered species; nobody is being hurt if the server automatically adds the prefix.
When does the prefix matter for end users? On print advertisements, TV and other offline media. In this case you need something to communicate to the user that he's seeing a web address. If you're not promoting "www.", then that something is "http://", which certainly isn't shorter or prettier than "www.".
When does the prefix matter for the technology side?
Use "www." or not, as you prefer. Just own up to the fact that it was a matter of person taste, and not a choice dictated by usability, efficiency, or technological considerations. And set up 301 redirects to the right domain.
According to pingdom, there were 234 million websites online in 2009, so it would be fair to say that if each website gets hit at least once, you'd save close to a billion keystrokes.
The more relevant point for webmasters, I think, is to support both variants (www. and not), but ensure that the website is set up to 301-redirect to one or the other to maintain a single canonical address for the benefit of search engine bots (SEO).
Earlier this year, a blog called ReadWriteWeb wrote a post about Facebook's login process. Because it's a popular blog, it soon appeared near the top for the search query, "Facebook login". Facebook users began to confuse ReadWriteWeb with Facebook and became angry when they weren't able to login. The ensuing comments are priceless and I really wouldn't do it justice to try to describe them, so I'll just link to the post.
A few days later, UX Magazine wrote a post covering the incident. In it, they also shared a video in which people on the street were asked what a browser is. If you've ever enjoyed watching Jay Leno's Jaywalking segment, you'll enjoy the video. If you make websites for a living (or browsers), you'll probably laugh nervously for a moment and then sob uncontrollably.
The point is, as webmasters, we really need to support as many different ways to access our sites as possible. Technology is a huge hurdle for many people and they do their best to scrape by, but for most, they're lucky if they can find what they're looking for. Even the supposedly tech savvy younger generation doesn't fare so well. Would it make sense to get rid of the www altogether? Probably, but it would also cause a lot of confusion and so it's important to support both.
In general, anything that you can do to make it easier on the user, you should do. Users aren't dumb, but I think we can sometimes fool ourselves into believing that our field and tech skills are shared by more people than reality would seem to indicate.