I too fall into the old as dirt category, so I have a lot of PC games that are rare by virtue of becoming abandonware. So I'll start with consoles...
Dreamcast:
Typing of the Dead (is this really that rare?)
Urban Chaos
that Dragonriders of Pern game
GBA:
Atari Anniversary Advanced (no particular merit to its rarity, it's just that they didn't seem to press a lot of copies)
PS1:
Um Jammer Lammy
PS2:
Rez
Ico (2nd pressing that they did when SotC came out)
XIII (is that really rare too?)
Midway Arcade Treasures 1&2 (again, no particular merit here, just that they seemed to have really small print runs, especially when compared to #3)
PC:
King of Dragon Pass (not rare, as much as part of a really tiny market, and still readily available)
Interstate '82 (sequel to Interstate '76, which I also own)
Darkseed
Weird Dreams
Fool's Errand (the Amiga version, if I recall)
Nuclear War (and I yet, I still haven't purchased Defcon)
Chris Crawford's Battle of the Bulge
V for Victory
Megatraveller
Twilight 2000
the Battletech RPGs
some of the older Sim titles, SimEarth, SimAnt, maybe SimFarm...
and yes, Alice