SmarterStats
Thanks to a recommendation from Scott Watermasysk, the father of .Text, I just downloaded a web server log analyzer called SmarterStats, and gave it a try. This is a sharp tool, built with ASP.NET, and very useful, from the little I've played with it. You can try the online demo to get a feel for the kind of information you can get from your logs.
An important selling point, in my mind, is that a single-site version of the product is available for free. So you can evaluate the full feature set, or run the product on a single site, for as long as you want. No time-bomb, no cripple-ware. And unlike other stats products I've tried, this seems to be a pretty intuitive and easy-to-use tool.
I haven't decided yet whether I'm willing to drop the $200 for a 50-site license (I'm leaning towards it, though I wish they had a 10-site license that was a bit cheaper, since I have nowhere near 50 sites to track stats on), but the free 1-site license is a no-brainer. Sorry if I sound like a commercial for this vendor, but it's rare when I come across a product that impresses me this way, so I wanted to share.