I can't say I "collect" fanzines, but I do seem to have a lot around. But compared to my (hard copy) comics collection it pales in comparison.
I recently bought a bunch of back issues of Alter Ego on eBay (all GA-related material, thanks to this site's influence on my feeble mind), I have some Squa-Tront, some DC Comics Amazing World issues...Various Artist tributes that seem fanzine-like--Kirby, Cole, Fine, Frazetta. Comic Book Marketplace, something called Third Rail (1981), Amazing Heroes, Buried Treasure, Comics Feature, Comics Journal, and a ton of Comics Interview--man, I loved that mag---read it cover to cover every month no matter who was featured--artists, inkers, colorists, letterers. Eventually became a letterer myself. I was so into all the behind-the-scenes info.
Moondood