If you are serving static contents (either html, or html generated by something like moveable type) then the answer is, for any contemporary hardware, a lot. Where a lot is probably measured in tens of millions of static requests per day.
In that kind of setup, the first limitation you'll run into will be the size of your servers connection to your data center. Most dedicated server vendors will start you off with a 10mbit connection, which is probably the first thing that will max out if you approach the number of requests quoted above. Generally they will switch you to a 100mbit port for little or no change, but be aware this just means you have a 10x increase in how quickly your bandwidth cap (if you have one) can be exhausted. Pay close attention and monitor your monthly useage closely, lest you pay large overage fees.
So, once you've got a 100mbit connection the next potential problem will be the speed of getting your data from the hard drive to the network. Even at 100mbit that is still only 12mb per second from the hard drive, which is trivial for contemporary hardware. Given a decent amount of free memory (for disk cache) and a good mix of file sizes (from a few hundred bytes for your favicon.ico to a few hundred kb for a big photo) you'll still probably cap out a 100mbit connection before hitting serious load.
However, all of this is assuming a site that serves static content, which is almost never true. If you're using a web framework like Django, Rails, Grails or any of the hundreds out there then your first bottlekneck will be CPU, the second will be memory, and the third will be the amount of concurrency your application can handle.