I guess I'll do this by system in descending order of rarity. Turns out I've got quite a few, though none of them are exceptionally valuable. I'm limiting things to the top 50 rarest per system according to the OP's link.
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Mega Drive: Streets of Rage and Ultimate Mortal Kombat 3
Sega CD: Earthworm Jim: Special Edition, Amazing Spider-Man vs. The Kingpin and Sonic CD
Dreamcast: Shenmue and Sonic Adventure
NES: Tetris
SNES: Super Mario RPG and Chrono Trigger
N64: Conker's Bad Fur Day, Star Fox 64 with Rumble Pack and LoZ: Ocarina of Time
Gamecube: Beyond Good & Evil, Skies of Arcadia Legends, LoZ: Wind Waker, LoZ: Twilight Princess, Super Mario Sunshine and Tales of Symphonia
GBA: Sonic Battle and Metroid: Zero Mission
PS1: Final Fantasy VII
PS2: Final Fantasy X, God of War, Kingdom Hearts, Shadow Hearts and Silent Hill 2
Xbox: no listing available
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I don't know which one out of all these games is actually the rarest, but Conker seems to be the most valuable. Then there's also my PSP, DS, PS3 and 360, but nothing I have for them could be considered in any way rare. I do question the sites prerequisites for rarity, since quite a few of the supposedly rare games I own are uncommon at most, or at the very least not hard to find to anyone keeping an eye out.