My first game here was Beyond Good & Evil(i think it somehow also was the reason i found GOG) and since it is a pretty short, straight-forward but VERY captivating game, i finished it straight away

might and magic collection

Not counting the free ones that were available when I joined...
FlatOut
Neighbours from Hell Compilation
And yes, I've completed both.

Fallout 1 and 2 (got both for 1 price, a promotion for the beta of GOG). Haven't finished any of those (yet).

Duke Nukem Manhattan Project - ended in a middle?

My very first GOG purchase was MDK 1+2, followed by Giants: Citizen Kabuto two days later.
I finished MDK back when it was new, though not this time around. MDK 2 I've only played very little. I haven't finished Giants yet. I'm into the Kabuto missions, but other stuff came up, and I never got around to finishing it. Maybe I should do so soon.

That was a while ago, but I think my first was Arx Fatalis. If memory serves I was still a bit skeptical of GOG at the time, and thought it would be a good game to test out the service with as I knew the original version of the game tended to have problems with modern systems. Many games later I can say that it passed my test quite nicely.

My first was Gothic, I think. What a deal. :)

Nice thread.
Fallout, and yeah, I beat it quite happily. My co-worker wouldnt shutup about it and I knew this new service had it for cheap so I gave it a shot. I remember being annoyed with it at first (this was the first game of its kind I'd played on PC)...clicking this or that wouldn't give me the results I wanted. But I figured it out after a couple hours, started over, loved it. When I got the leather jacket at the start I became addicted. :)
Character was a lady, master of hand to hand combat...real mean with the Power Fist.
I remember being amazed at all the free stuff that came with it, especially the Fallout bible.
Shortly after I got Fallout 2, then awhile after I guess the free GOGs got me back into it. Was addicted from there on.

Besides the three free games, my first buy was during the Kalypso discount:
Commandos Ammo Pack, Commandos 2+3, Tropico Reloaded, Patrician 1+2, Praetorians,
Imperial Glory and Port Royale, with a $25 discount. Not bad, huh?

First i grab the Free games to check out GoG's Handiwork, then i used a Promo Code that i got with a Magazine i subscribed to for my choice of Three Games - i chose Kingpin, then i got a Gift code for Tex Murphy Under a Killing Moon (Thanks to Interactive Distractions) and finally my First Purchase was the Prince of Persia Sands of Time and Beyond Good and Evil in the 2 For 1 Offer
No i Have not Yet Finished any of them, though i have completed some of my later Purchases before Getting them on GoG

My first was Operation Flashpoint. Good game, but I find it to be very difficult and I never made it all that far. I really should get into it again.

Freespace 2 with Descent 3 as my free game

Colin McRae Rally 2005 and Screamer (from a promotion with them both).
It took me forever but I finished CM. Screamer, on the other side, I couldn't finish.