Mine was Kirby's Dream Land on the original Game Boy... though I had a Game Boy Pocket... that's beside the point. Yeah, one of my first handheld titles, and as far as I know, the first game I ever actually beat. Wonder whatever happened to my copy... I lost it years ago...
My first one was Nightmare in Dreamland. I already knew OF Kirby, and I just wanted to get one of his games. I got tired of it pretty quick though. Much more fun playing 64 and CC.
NiDL was my first. I had heard of Kirby too, exactly like what DG said. I wanted to pick up Tilt n' Tumble, but I couldn't find it anywhere. The came KatAM, then Air Ride, then the DS games.