Well, I'm a few days late, but I guess better late than never!
I got a Game Boy for Christmas as a child shortly after it was released, and I definitely spent a significant amount of time gaming on it when I was little. There were a lot of long car trips filled with Tetris, Mario Land, and whatever garbage licensed games I wound up with over time. That being said, while I definitely spent a lot of time playing Game Boy games when I was younger, I've always felt somewhat cool towards it as a system, probably owing to some of the things Maru mentioned in his post. While I love me some pixel graphics, a lot of the old Game Boy games, with their monochrome palette and blurry animation, just look ugly to me. Even games that I genuinely love, like Link's Awakening (in my top three Zelda's) and the Wario Land series, have what I would consider only serviceable visuals. And a lot of those early titles were very, very simple as well, feeling like they were just stripped-down versions of console titles. They were good for when you were traveling, but as soon as you were home and had access to superior console games, why bother playing a Game Boy?
In the last year or so, though, I feel like I've started to warm up towards it a bit more. Later titles for the system are typically significantly more robust than those early releases, and I certainly won't deny that some of the Game Boy Color games look and play genuinely fantastic. Like some other folks have said, Dragon Quest/Warrior III is an excellent port, and the three Zelda games are absolutely among the series's best. While it was talked about a fair bit at the time of their release, I feel like the linking (pun intended?) aspect of the two Oracle games has been somewhat lost over time. The two games are designed to be played back-to-back, with a password to carry over some of your stats and character relationships. But unlike some other games that have some degree of connectivity between titles (Mass Effect, Quest for Glory, The Witcher, etc.), it actually doesn't matter what order you play the games in. Whichever one you play second will have some additional scenes and content that connect it to the game you played first, including an additional final section where you confront the true antagonist who has orchestrated the events of both games. It's really neat, and it's something that I always enjoy each time I replay the pair.
At present I'm hoping to be able to get my hands on a copy of The Fish Files. It looks absolutely incredible for a GBC game, and I've heard it really nails the tone of old Lucasarts point-and-click adventure games.