Is there any way to prevent logging (hosting services/scripts)?

I was wondering today, every single host, free or paid (who supports more than HTML) have an option in their cpanel to log not only the owner of that account (his IP), but everyone who have accessed his personal website.

And some other kinds os statistics, for instance, referrers, useragent, how many time did the users spend on that page, etc.

I am asking two different questions - is there any way to sign up a host (free or paid) which has a Control Panel with good options, but is 99,9% modified in terms of privacy? Who doesn't log or register anything regarding other users (if not the owner of that account)?

And what about these scripts developed today, like phpBB, Invision, vbulettin and others, is there any way to prevent them from log any IP from the users? A MOD can do the job? To remove completely this option from log/register any IPs, even from the admin itself?

I know this second question should be asked to several support forums, but it's a general question for everyone of them. What I want is a host who might even log my access, but not log anyone else (this should be my Privacy Policy).

Just for the record: The Cpanel from free/paid host services can't be modified by the users.

 

Another useful information:

I don't know about the others, but phpBB 2 has one useful MOD which not register any IPs from users:

http://www.phpbbhacks.com/download/457

Disable User IP Storing

This hack will make your board operate without knowing any or ever recording any user ip addresses.

All IPs are being recorded as 0.0.0.0.0 instead.

Still remains the question regarding the host services.