I am going to focus mainly on exclusive games here.
Sonic Triple Trouble is the only 8-bit Sonic game not also available on Master System, and it happens to be the best of them all. Very good graphics for the system, and great gameplay. Even the music is good.
Defenders of Oasis is a terrific Game Gear exclusive RPG, and unlike most GG RPGs, was actually released in English. Why they bothered with Japan-only RPGs on a system that bombed in Japan remains an irritating mystery to me.
Speaking of RPGs, Crystal Warriors is an incredibly good strategy-RPG. It's better than the GG Shining Force game. Really well-made, with a great battle system.
RC Grand Prix is the system's only exclusive racer. As you can surmise from the title, it's along the same lines as RC Pro-Am.
Both GG Shinobi games are completely unique to the system, and are awesome. Kind of along the same lines, there is a Ninja Gaiden for Game Gear which is also completely different from any other Ninja Gaiden (including the SMS one).
Halley Wars is a surprisingly good vertical shooter. The GG Aleste games are also good.
Super Columns might be the definitive version of classic Columns gameplay (Columns 3 changed it around completely). For some reason it was only on Game Gear.
Game Gear was home to 3 exclusive X-Men games. The best is the second, subtitled Game Master's Legacy. But all three are worth playing.
Pengo is a great port of an arcade game from 1982.
Both Jurassic Park and The Lost World are rather good for GG, and different from all the other games with the same title. The first Jurassic Park might be the same for Master System, but I've never played the SMS version. The Lost World was the final GG release worldwide.
Arena: The Maze of Death is an interesting case. Released later in the system's lifespan (1995), this game was created from the ground-up for Game Gear, and doesn't exist on any other system, and it's not based on any other IP either. It's a unique action-adventure game with an isometric perspective.
GG-exclusive rip-offs:
If you want to see a weird attempt at ripping off Metroid, Tails Adventure fits the bill. It's more linear than Metroid, but has the same basic gameplay elements. There's a good rip-off of Cadash in the form of Ax Battler (in which you wield a sword, but whatever). Woody Pop is the best Arkanoid rip-off ever made.
16-bit Ports:
At the time they sold the system, but in this day and age, there's really not any reason to bother with GG ports of Genesis games. With a couple of exceptions. Ristar is worth it just because of how good of a port it is. Dynamite Headdy is worth it because it has differences in levels and gameplay (as well as making the transition rather well). Lemmings might be worth it, as it's pretty much just as good as any version. Urban Strike is interesting because of how "advanced" of a game it is. Super Return of the Jedi is kind of interesting for being a port of a 16-bit game that was only available on SNES. Galaga '91 is the system's only TG16 port. And Baku Baku is a Saturn port that somehow works out well.
SMS Ports:
Generally speaking, I would recommend SMS versions over GG versions when available. However, if you don't have an SMS or Power Base Converter and have no intentions of getting either, the GG versions are usually fine. Here's a few games from that category I would recommend. First off, Wonderboy: The Dragon's Trap is an incredible game (also available for TG16 as Dragon's Curse). Secondly, Vampire: Master of Darkness (aka In the Wake of Vampire) takes a page from Castlevania but is one of the best 8-bit action games ever. Third, Bram Stoker's Dracula is a great fast-paced action game, and actually much better than the 16-bit versions (which were a different game). There's also Revenge of Drancon, a weirdly-titled port of the original Wonderboy; Fantasy Zone, the classic shooter; Sonic 1, 2, and Chaos; and Dragon Crystal, a very odd mix of RPG and puzzle. GG didn't have all that many SMS ports considering the hardware (I guess because of the Master Gear converter).
Edit: Nevermind, I made this post before it was merged into the Top 20 thread.