It all depends on what you are looking to extract from your analytics, the right tool changes according to the metrics that are most important for you. I'll put some of my favorites and the reasons why and when I use each of them:
This is a problem of understanding if you have a screw or a nail, then choosing the right tool for it. Hope this helped.
Google Analytics is free. StatCounter has a free model and paid one.
xiti (now 'at internet') offers good paid analytics tools. It is originally a French company, now with customers from Canada and UK (as far as I know).
I am not a direct customer, but I am a user (our ad network is paying for us) and the service is great and the numbers seems to be closer to reality than with other analytics tool.
Depends precisely on the business requirement for analytics information during the startup phase.
Do you just need a general overview ? Then google or piwik are free and useful.
Do you need something highly specific such as realtime geolocation and mobile client types to make day-to-day business decisions ? Then you need to look at the specifics of each offering.
The best answer for your specific needs might be paid, free or internal dev or some blend thereof.