First game was Fallout 1 & 2, around the time GoG opened its doors.
They were truely hard to find at the time, and GoG was the only way to procure these kind of games which had completely disappeared from the stores (physical, we weren't there yet for the digital ones), which everybody knew, talked about, but many players had missed because too young at the time they released.
The actual first game on my library at GoG is Tyrian 2000 followed by Kid Agent but at the time them and a few others were givaway for registering or buying your first actual game.