I agree I use Google analyitcs daily for my site, it gives me more information than previous stats programs and costs nothing.
I suggest using multiple and only using those which do not insist on a button on your homepage. I use Analytics, webalizer and awstats; together they give a nice overview; they all have shortcomings and strengths. In general analytics is the best, but it is not an 'exact' science what a visit, a hit, length of a visit are; they are subject to the analytics package's programmer algorithms / heuristics. Also; Google An is limited for a lot of purposes like tracking exact income from multiple sources (you can do a lot with Goals though) or integration with, for instance,social networks; when you track, for instance, your Facebook app you have *a lot* more information about people than Google will use to create the stats. Using multiple free systems is, imho, the best way.