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 logging side of this is either PHP code that logs directly or Javascript that waits two seconds before logging in order to filter out real users from some bots (and I'm not interested in immediate bounces). The Javascript option is useful for static websites (like this blog). The logs are periodically purged.

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.