As Lix stated, really the only way, without storing IP addresses or forcing users to register, would be to store a Cookie linked to a session, however of course this is by no means a solution, as Cookies can easily be cleared.
You could attempt to use a persistent Cookie solution, such as evercookie, but if I were a user of a site which implemented something like that, it would annoy me more than having to register, and no matter what methods are used, they can always be circumvented to remove the Cookie/identifying information.
Another possibility is another form of persistent Cookies which got a lot of publicity last year (or was it 2012 - I forget), which utilize Flash to store Cookies. These Cookies can be accessed in multiple browsers. One such solution is flash-cookie, but again, this is likely to annoy users, and would not be 100% reliable as not all users have Flash enabled or installed.