GG Aleste 1 & 2 are amazing. I also really liked Sonic 1-- I enjoyed that one more than 2. It was so good that I wanted to play it through again immediately after the end credits rolled. Let's see.. there's a great version of Mappy, Dragon Crystal is a very nice roguelike, Halley Wars is good, there are incredible ports of Gunstar Heroes and Ristar, and the Puyo Puyo games are fun as always. You might also check out Popils, which is a fun puzzle-platformer. Sonic Triple Trouble is also quite impressive but suffers just a bit from trying to cram those 16-bit physics and play mechanics onto the GG, but it's worth trying to see if you like it. And there's Galaga '88-- I mostly played it on Turbografx, but that's an awesome game and I bet the GG one is pretty good. Someone mentioned Road Rash, and that is an incredible port. That and the SMS version both knocked my socks off. If you're coming off of the Lynx, you might be right at home with it. That GG Fantasy Zone is pretty cool, too. And Factory Panic is a clever one.
Guys, we don't have to argue Game Gear vs. Lynx. They're both great handhelds that are good at different things. Like as much as I played Space Harrier on GG (probably my most-played GG game back in the day), it probably would've been better on Lynx. And Lynx is probably not going to be able deliver GG Aleste 2, just going off what I've seen from the released Lynx games, because it absolutely needs a high refresh rate. Get them both, love them both .... and keep some spare capacitors handy.
Add the GameBoy in and you have 3 really distinct handhelds: The Lynx had a lot of arcade and computer-game-style games (simulators, into-the-screen scrolling games), the GG was all about console-style games (platformers and shmups), and the GameBoy is a treasure trove of great puzzle games.