I've been using Google Analytics, but really would like software that provides real-time data, in addition to providing more detail about users that Google doesn't provide because of "privacy concerns"

I found Clicky web analytics today and it looks pretty good. Here's their comparison chart:

http://getclicky.com/33034#theotherguys

Has anyone used their service?

Any recommendations as to what service might be better then theirs, in the same price range?

AwStats and Webablizer Stats with Fantasico are all I ever need.

Getclicky is good, I have been using them for over a year

There are many great wordpress plugins I could recommend as well if your site is a WP powered blog?

Yep. Awstats is very good for analyze our visitor..and for others i prefer to sitemeter and it's good enough for me.