If you want to track how many times users revisit the site, you could do that anonymously by setting a visit counter cookie, e.g. VISITS: 1, VISITS: 2, etc. This would track the user over different IPs, but since the cookie only has a counter, it doesn't tell you if two people with "VISITS: 2" set is the same user.
That's the first example I can think of off the top of my head.
This sounds weird, or wrong. Does anonymous stats need cookies at all?