Probably no games, since I prefer to collect for disk-based systems. About the rarest thing I have according to the database is my complete copy of Pipo Saru 2001, which was a Japan-only Ape Escape spin-off game. I'm one of 4 people with that in their collection. Oh, I'm also one of two people with the Japanese version of Devil May Cry.
As for swag, I don't have that in the db but I imagine I have quite a bit that's unique or at least rare. A Steambot Chronicles mini harmonica keychain, a Starfox Adventures car air freshener, a GBA SP neon wall clock and my double-sided Legend of Dragoon promo poster, just to name a few off the top of my head. I want to do some videos at some point of me going through some of my stuff like that, because I find a lot of it interesting so maybe other people would too.
EDIT: Oh, and this is my wedding band - it's probably somewhat unique.
EDIT EDIT: Ack, I completely forgot I have the Killzone 3 Limited Helghast Edition - probably because I need to get it entered into my collection.
The Killzone 3 Limited Helghast Edition[28] is not sold in stores and is exclusive to North America. Only 1000 have been made, 125 of which were given away in the multiplayer open beta from February 4 through February 6 and February 11 through February 13, 2011. The Limited Helghast Edition contains all the same elements as the retail version of the Helghast Edition, but features a replica Helghast Helmet with illuminated LED eyes, simulating Helghasts’ glowing mask.
The outer box of mine is signed by the guys from IGN's Podcast Beyond. Unfortunately, when they shipped it to me they just stuck a shipping label on the actual box, so it's a little beat up.