If HTTPS Everywhere still stores too much information for your taste, try Tor. The system was designed for the US Navy to protect government communications, but now is available for use by the public. It’s a free download that makes all of your online visits anonymous. The downsides are that it slows down your browser.
If you want insurance of browsing security, nothing is full-proof, but Tor comes pretty close to it.
Check out this inforgraphic from the Electronic Frontier Foundation that shows the differences between using Tor and HTTPS.
