So is Complete Anonymous Web Surfing Possible ?
The main problem is that to surf anonymously you need to protect various elements of your browsing and generally people only protect one. Here's a nice little diagram showing your average browsing session.
The main thing to notice and without this knowledge any attempts at anonymous web surfing are doomed to failure is the simple fact that virtually all browsing is conducted in clear text, that is everything is easily readable - account names, web sites all the information you pass to the web server and vice versa. This of course means that all this information is vulnerable as it passes across the wire.
Take your ISP for example, every piece of data is logged by them and stored on backup tapes (in Europe there is a directive which states they should be kept for two years)
On those backup logs is a list of just about everything you have done online for the last two years - this is incredibly powerful personal information - do you know how your ISP look after this ?
So complete anonymous surfing has to take this in to account - the standard solution usually suggested is to point your browser at an anonymous proxy and you'll be safe, however this is completely misleading and in many cases extremely dangerous. Remember you get nothing for free in this world and lots of the free anonymous proxies are in fact run by identity thieves on hacked servers. You won't get anonymous web surfing just your personal data stolen.
More Anonymous Surf stuff to read
Read some more here - Proxy Surfing