Maybe a broad question, but how can you keep a visitor to your site 100% anonymous on your server. The site is written in PHP.
No trace of what he or she has visited on your site. Any suggestions?
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One main thing is going to be disable or limit access logging. You would have to configure your Web server not to log any user's IP addresses.
Also ensure access is not logged elsewhere in the system or the data center, no external applicances like reverse proxies are active, and so on.
Also as @JonahBron points out, don't use sessions. Best don't serve any cookies at all.
Encourage your users to use additional anonymization services on client side, like TOR.
Related interesting reading: Wikileaks' Hosting
If you are concerned that your application's visitor data may fall into the wrong hands, you should never collect any data an attacker could use against your visitors: this means your application should not be publicly available, because an attacker who gains access to your server or the network your server hosts on will always be able to monitor incoming and outgoing connections.
Make your application available only via TOR hidden service if you care about your visitors' privacy.