I use Google Analytics on my site...
But, I would like to display some traffic info to the users, like geo localization or just a hit-counter on the main page of my site.
Does Google offer something like this?
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Yes, you can through the Analytics API. For an example of a site that uses the API to openly display data from Analytics including Page Views, Time on Site, Direct Referral, Search Referrals, and more see this SEOmoz post, and then click "Post Analytics" at the bottom of this or any other blog post on that site.
However, note that it sounds like the data can be slow to retrieve, and there may be other limitations, so you may not want to use it in a traditional 'hit counter' fashion. (I would search StackOverflow for the many existing topics on simpler hit counter methods.)