1) I use Google Analytics on every website... but I always find myself mildly unimpressed. The UI is pretty, but is too busy and complicated. Most importantly, it isn't real-time. And in the world of search traffic, linking, etc - real-time is critical. Analytics is unique, however, in two important capacities: being able to set activity goals and being able to integrate with AdSense. Unless you are using expensive systems like Mediaplex or Omniture - action-based tracking is quite valuable.
2) I install the wordpress plugin SlimStat on all of my wordpress-based sites (which is more than traditional blogs). SlimStat is terrific because it is shows every incoming click, referral, domain, etc. It is fascinating. I can press refresh and entertain myself for hours. It also is immensely helpful (particularly for SEO experts and/or serious bloggers). SlimStat has some serious problems though: it doesn't scale well at all. Under 25,000 pageviews a day and it hums... Anything more and it is a memory suck that takes seconds to load (suddenly making real-time not so real-time). SlimStat also doesn't archive data. So it's a great snapshot of what's happening now - but not what's happened overtime.
3) I turn Quantcast on for all of my sites. In terms of basic traffic statistics, Quantcast makes things remarkably simple, enables easy comparisons between sites, and gives an accurate third-party / public representation of your data (which can be quite important).
I've never personally paid for an analytics application because I haven't had a need to. Between Analytics, SlimStat and Quantcast - all of my needs have been served.
But for some reason I paid $30 for Mint last night and am thankful I did. Although I'm not going to turn off those three other accounts, I will be checking Google and SlimStat with far less regularity. Mint does 95% of the total job - but it does that portion extremely well. And the best part - it's customizable, lightweight and the code is flexible enough to enable third-party apps and plugins.
I'm addicted. Already encouraged numerous people to give it a try.