Yeah, so I’ve been itching to play a good RPG on my Gameboy lately. I’ve got:
FF Legend 1-3
FF Adventure
Pokemon R/B/Y/G/S(J)
Megaman Battle Network 3 White
DBZ Legacy of Goku 1&2
FF1&2: Dawn of Souls
FFT Advance,
Zone of the Enders: Fist of Mars
KH: Chain of Memories.
Any and all suggestions would be appreciated 
The Golden Sun games are pretty awesome for a GBA specific RPG. I’m also playing Dragon Warrior Monsters (i think thats the title) for the original GameBoy, if you can play those games on your system.
I like the GBC version of Dragon Warrior 3, if you can avoid the crippling, game-breaking glitch that ruined it for me (though I’m not sure what triggers it).
Mario & Luigi: Superstar Saga is awesome. Get it just for Fawful.
Riviera: The Promised Land - Has a few non-standard RPG aspects to it (No free movement of character) and may seems like something from a harem anime (1 guy with 4 girls) but still a very solid game and played it through 3 times. Might be hard to find, being published by Atlus.
Mario & Luigi: Superstar Saga - Fun little game that has the Mario RPG feel to it.
Fire Emblem - Haven’t actually played this series for myself but always seem to get suggested and set the standard for portable SRPGs.
Tactics Ogre: Knight of Lodis is an excellent GBA TRPG, and the same goes for the Fire Emblem games.
All non-fog stages in the Fire Emblem games, at least.
Also, don’t play the Fire Emblem games on an emulator unless you want to ruin them.
I’ll join in the M&L crowd. This is the most fun game for GBA I’ve played. First party developers, yep.
Now, not that I’m a fan (gross understatement) but does the word Zelda ring a bell?
Lets see. If you haven’t played Zelda:A Link To The Past for SNES, you may get it in its GBA version, which is almost the same and has an extra dungeon or sth (multiplayer only I think). GBA isn’t such a hit where I live, so no specifics on that. You’ve played that, right?
Link’s Awakening (DX if you want colour) is a game worthy of the zelda mantle. It took advantage of the original GB’s puny hardware and made a thoroughly enjoyable game. I prefer it white and grey, but colour is in the eye of the beholder.
There were the two Oracle titles which seemed like fun, but must be finished in one big playthrough or you forget what you meant to do where. The experience seems a bit close to Link’s Awakening.
Then there is the GBA Zelda: The Minnish Cup. It is fun but more heavily quest-oriented (the exploration part has been cut down a lot, shame on you Capcom). The various quest parts take up a lot of time (otherwise the game is a bit short). Then again you get some rewards and the various characters lend, well, character to the main city. And Link can do tricks, hah. There are a few good items and a few lacklustre ones. Enjoyable overall, yet not amazing.
There are 3 Dragon Warrior GBC titles fi you can stand them (I cannot).
There are a few remakes for GBA if you are keen on that (Phantasy Stars and Breath of Fires. I won’t rant about BoF now)
There is Shining Force: Resurrection of the Dark Dragon which is Shining Force I with better graphics and I will surely replay this (some day, that is). Tactical RPG, you probably know it. It was Buddha’s gift to Mega Drive.
Last calls: if you are not action averse (and DBZ suggests you aren’t) get Castlevania:Aria of Sorrow. It single-handedly changed my opinion of the series (and made me play some more). Action with a lot of RPG elements (levels and magic and inventory etc).
I’ll just add metroid which has absolutely nothing to do with RPGs but is a good game. And Tetris, hah, the soul stealer.
(Does it show that I just woke up and haven’t seen what I’ve written? No preview for this)
FF4 is coming out for GBA i think
FF3, 5 and 6 are coming out for DS… 
I highly recommend Golden Sun and GS: The Lost Age. Really fun puzzles. Also, the Castlevania games have a lot of RPG elements, though not really RPGs. Megaman Battle Network games are catchy, though extremely derivative.
Like TD said, you can also try Tactics Ogre: Knight of Lodis and Fire Emblem.
ARGH, that hurt. Straight to the heart.
edit - Wasn’t it Pachisi somehow?
How could you people forget Tales of Phantasia? For shame…
ToP is an SNES game. Or a PSX game. It’s only on the GBA because Namco is greedy. >_<
Holy effing shit! FF5? Portable? YES! THANK YOU, MY VARIOUS GODLIKE THINGIES!
And yes, Fire Emblem is always a good one… I own the three that are American-released. Extremely customizable characters, and some of the best stories out there. Seriously. Get Tales of Phantasia too, but only if you have a SP or DS. The graphics are awfully dark in caves and dungeons, and I seem to find myself having endless trouble shifting about to get the best lighting.
…extremely customisable? From what I know, they’re basically locked into a class (and an upgraded version of it) for life. Possibly with the later releases having branching paths, but I’m not sure of that.
Regardless, extremely customisable isn’t what I’d call Fire Emblem characters.
Pokemon Fire Red/Leaf Green deserve mention. They’re good for a nostalgia trip. They’re the first generation Pokemon games dandied up to look like Ruby and Saphire.
FF4 is out already. I played through it - it’s a damn fine game. There’s a wierd little speed issue that lets you get in about twice the attacks you could in the original, so it’s actually the easiest version of the game to date. The added stuff (two dungeons, many bosses) is fairly fun, since the larger of the two extra dungeons actually gives you a bit of new story for each character.
I’ve really been considering Golden Sun, Fire Emblem, Fire Red and Green Leaf. The FF Games on the DS is something I’m already looking forward to as well. 
I’m rather curious about what Tactics Ogre is about though. Would someone care to tell me a bit about the story without spoiling?
Also, how does Sword of Mana compare to Final Fantasy Adventure (Seiken Densetsu)?
Apparently Sword of Mana’s story is very similar to FFA (I haven’t played FFA myself) but the gameplay is a mix of Legend of Mana and SD3.
Gifted soldier goes on a mission with longtime buddy/bigwig paladin friend in a country which they brutally took control over. They get seperated, he starts to have doubts about the mission, his nation, war in general, etc. Pretty decent plot, with a bigass branch in the middle, so good replay value.
FireRed/LeafGreen are extremely good plays. I was talking abut The Sacred Stones earlier, btw. My bad. But if you really want customizable, look no further than Pokemon. Seriously. They may be cheezier than a year-old toe infection, but they can be challenging if you don’t know what you’re doing, and you can always change your team in almost any way, shape or form. Almost. There are a few drawbacks, though. Some things just aren’t possible, for instance, having a Fire/Water type. Which would pwn. And some pokemon are really only good for one thing and one thing only, which leaves no room for versatility. Which sucks.
Some of the more notable GBA RPGs
Breath of Fire
Breath of Fire
Final Fantasy I & II Advance
Final Fantasy I & II: Dawn of Souls
Final Fantasy IV Advance
Final Fantasy V Advance
Final Fantasy VI Advance
Fire Emblem: The Sacred Stones
Golden Sun
Golden Sun: The Lost Age
Harvest Moon: Friends of Mineral Town
Harvest Moon: More Friends of Mineral Town
Legend of Zelda: A Link to the Past, The
Lufia: The Ruins of Lore
Lunar Legend
Mario & Luigi RPG
Mario & Luigi: Superstar Saga
Pokemon Emerald
Pokémon Fire Red
Pokémon Leaf Green
Pokemon Ruby Version
Pokemon Sapphire Version