I am possessive of... food. I don't like sharing that, it's kind of obnoxious when someone wants a bite of something, or mooches off me because they "aren't hungry". I am possessive of my computer(s), mp3 player(s), and my phone not because they have sentimental value for me, but because I don't want people looking through my personal things, texts, music, etc..
Also, I am possessive of my chihuahua, I think. That's only because I love her, though, and she is infinitely cute, and at night I want another living being with me in case I space out too much or get scared whilst sleeping in the downstairs room. She can't complain about how late I stay up, either.
I think only maybe with my computer.
People sometimes get frustrated with me because I don't get possessive of them and I don't get jealous. It's like they think I don't love them or something.
That is pretty funny, but pretty sad at the same time. I have that problem too, sometimes.
I used to get possessive of my friends, but as time goes on I seem to care less and less, as I find that they really aren't worth my effort: they don't care as much about me as I do them, and to add, they aren't particularly insightful or important to my life.