i agree that this is basically impossible with HTTP being stateless and all that, but you might be able to achieve something similar doing the following:
every page could have an embedded cgi somewhere (via SSI, or maybe a cgi-generated image), and that CGI-script could check for some cookie you set the first time the user hits your website. using the information in the cookie, you could then log the the visitor's 'tracking number' alongside the url and the time somewhere.

i'd be very surprised if somebody hasn't done this before, though...maybe something in combination with the various session-management modules?