Even if I may run into the danger of dating myself (and kinda feeling very old now looking at those other entries), in no particular order and of the top of my head the games I played before their predecessors:
Dune 2 (kinda does not really count, as the first one was a very different game: Adventure->RTS)
Zelda 2 (although I have to admit I started falling in love with the series only after A Link to the Past)
TIE Fighter(The DOS-era game, one of the few where I really enjoyed playing the "bad" guys)
Warcraft 2 (after that, I played all Blizzard Games chronologically)
Wing Commander: Privateer (which technically is a spin-off, but that was my first game of the Wing Commander series)
Turrican 2 (I am not entirely sure about this one, I might have started with the first one after all)
Duke Nukem 3D (at least I think so. Were the predecessor not 2D-sidescroll games? I am not sure here either)
There are even more hazy ones, like
SimCity (where I might have played 2000 slightly before the original) and
Metal Gear Solid (if you take Metal Gear into account). I also have trouble ordering the
Final Fantasy games, as I definitely did not start with the first one, but am not sure where I actually started. The picture gets clearer with VII and beyond (as for many people, I guess) and the harmonized launches and titles across the world.
Some of the more popular ones mentioned here do not count for me, for example
Fallout (played those) or Call of Duty (I played the first ones briefly but have actually kinda lost interest in most FPS games and so did not play the more recent ones). Otherwise, one of the few games from more recent memory where I skipped the predecessor (in contrast to skipping the entire series until now, there are lots of those

), I think the only one coming to mind right now is
Left 4 Dead 2. Odd, isn´t it?