Heh. I have a whole list of them that we use at work (but I do work for a web analytics agency, so that might explain why):
Google Analytics as thorpe mentioned, mostly realtime [they do have a new real time section, but for a majority of the reports there is a slight processing delay]
pion which integrates with other analytics tools, but allows you to replay visitor sessions
Piwik an open source of Google Analytics and is real time.
Omniture, not always real time (15-30min delays) but very powerful
Yahoo! Web Analytics is another real time one, but requires to either you to purchase Yahoo's PPC ads, use their ecommerce (not sure if this is still available) or have a consultant like us.
Stat Counter is a simpler one and tbh, I've never been a big fan of it.
Webtrends *shudder*, I can't believe I just added this one. It is a tool, and a powerful one, I just hate it Plus, 24hr reporting delay.
Each have their own pros and cons.
Why did I list analytics tools? TBH I don't really see the point in the "real time logging." Sure, you had 50 people using the contact form or 3,345 people are viewing this one page... but why? You don't see the whole picture, just bits and pieces of it. In other words, it's not actionable data.