I'm generally a believer that you get what you pay for. But with Google Analytics (GA) being free, you certainly get a lot more than you pay for.
I've been using GA on a variety of sites for over a year now. While I'm certainly no expert on it, I have gotten my hands dirty enough with it to appreciate all it has to offer. We recently launched a new pro selection service for Washington state at http://search.porch.com. In the ramp up of launching that site, I did an evaluation of the different web analytics offerings out there. I won't go into the details of what I found. But here's the summary of my findings.
You should just use Google Analytics until the time that you hit GA's traffic volume limits. If you have a need for capabilities beyond GA, then get a second package to cover that need.
Now Google Analytics isn't free for everyone. You do need to pay for it once your site exceeds 5 million PVs per month (a metric that I am looking forward to exceeding).
Here are some of the reasons that I think you should just use GA:
Now, not all is perfect. My biggest complaint about GA is about usability and learning curve. There's a lot of functionality here. But discovering it and mastering it can be a challenge. And there are some settings that I still have trouble remembering where they are in the UI.
Additionally, we had some problems implementing GA in our non-native iOS app. And I'm still not happy with our implementation there.
But the net of the conversation is that Google Analytics is a great tool for anyone that wants to understand how users engage with their site.