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"GoingUp! provides first class web site analytics and seo tools for your website. With GoingUp!, it's simple to track visitors, monitor sales, see conversion rates, and improve your search strategies. GoingUp! offers the most complete analytics around."

Offers two live web stats wigets - an arrival and departures traffic report and a global traffic map. Both require no registration - just copy 'n' paste the code, and you're done.

In their own words: "Performancing Metrics gives bloggers and smaller web sites a more personal understanding of their visitors. Many analyzers give good summaries, and Performancing Metrics is no different - but the similarities stop there. Performancing Metrics stands out with its refreshingly clean and simple interface, innovative features like Spy and RSS feeds, and an unrivaled per visitor level of detail. You also get real time stats, outbound link tracking, download tracking, IP tagging, custom data tracking, and much more."
Something else which should have been here a long time ago, fantastic and recently redesigned web traffic analysis for an unlimited number of websites... From Google.

Very cool (and free!) web traffic analysis, including a feature which lets you view the activity of users currently surfing your website. Currently only beta registration available and you have to wait for an invite - signup now!

Monitor your website's search engine rankings. A free plan gives you a watchlist of up to 10 websites.
A free yet reliable invisible web tracker, highly configurable hit counter and real-time detailed web stats.
103bees "is a real-time online tool for webmasters and bloggers that is highly focused on natural search engine traffic analytics. It provides tons of detailed statistics and in-depth information on the search terms that drive visitors to your webpages." A great tool if you want to 'get serious' with SEO.
"Learn more about your website. Live views, recent statistics, and lifetime summaries combine to form the most complete picture of who's using your site, what they're doing, and how they got there." Free accounts provide support for tracking just one website and no access to the API.

Measure Map provides free, informative and good-looking statistics analysis for your blog. Unfortunately no new memberships can be created at this time as Measure Map recently became part of Google and they are currently "making improvements". You can register your interest by entering your email address on the current site though and you'll be informed when free accounts are on offer again.