I don't find Google Analytics to be very insightful. A big disadvantage is that it's mostly not real-time so you lose out on time-sensitive opportunities/insights.
There is probably a nice analytics service I could buy but it's also fun to roll your own software. The screenshot below shows my personal logging system (click for a better image).
Each of my sites logs information which is then amalgamated by a PHP script that runs locally on my computer. Some of the logs have information about the currently logged in user (for the subscription services I run) so I can keep an eye out for when there are new users/people who might need help. The reverse DNS for each IP is shown so I can sometimes identify companies/organizations that are on my site.
The key feature of my system is that it's always up-to-date and can give me a heads-up on when someone has posted a link somewhere that's driving traffic.
This service doesn't look too fancy but it provides the insights I need. I rarely check Google Analytics now.
Update: a friend of mine pointed out that Heap is a great choice if you're under 25k visitors per month.