Hmm, not sure what my geekiest possession is. They all seem equally geeky to me, so instead I'll focus on the items that aren't as obviously geeky (in no particular order):
First is my pair of Good Eats cookbooks. Which may not sound geeky, but I assure you that Good Eats is one of the geekiest things out there. One of these books discusses the molecular structure of fudge, just to give you an idea of what kind of books these are. Alton Brown is awesome.
Second is my mandrel jig. It's basically two vertical pieces of wood set onto a 2x4, with a metal rod passing between them. It's used to wind wire into coils for making chainmail.
Third is a key holder with mine and my fiance's name written out with computer keys that are glued onto the wood.
I have a photography lightbox I made myself out of cardboard and tissue paper.
It's also possible that I own art pieces that have been in museums. My fiance's grandfather was an extremely prolific artist (EXTREMLY prolific--when he passed away everyone in the family took several paintings, carving, and powder-horns, and the house was STILL filled with them) and several of his pieces have been displayed at small art museums over the years. Seeing as I now have one of his finished horns (and another that was carved, but not inked, as well as a few 'blanks') and about a dozen of his paintings, it's entirely possible that I own something that has been in a museum. Even if not, though, I suppose owning carved powder horns is pretty geeky on it's own. (And the paintings are beautiful)