Mixpanel has a free tier for their engagement and people plans.

I'm only played around with Mixpanel a little bit. I'm a little more well versed in KISSmetrics but they don't have a free tier.

Mailchimp isn't an analytics tool but they offer analytics on their mail campaigns and they have a fairly large free tier.

Piwik is a nice self-hosted alternative to Google Analytics. We've used it for several projects and it works pretty well.