In no exact order (don't make me choose between muh babies!)...
Chrono Trigger - Question: You can't have one of these lists in the Compendium w/o CT, can you? Answer: No. We all know why it's amazing: great characters, really neat story, good gameplay, great replay value, etc. I won't bore people in the Compendium who already know how great it is.
Final Fantasy VI - At first known to me as Final Fantasy III on the SNES, I soon came to learn that in Japan, they had already had SIX games in the franchise of which this was, at that time, the last released. Well, anyways, as I'm sure others have pointed out, it's a great game. The characters were (mostly) all solid and well defined, each having unique, distinctive personalities. The story, while kind of cliche, is really well done & includes enough unique touches to keep the attention & drama up & pumping. Not to mention, it's got Uematsu's BEST work for the series. Amano did some great character designs as well. Absolutely amazing.
Final Fantasy Tactics - Square took two things and made them great: 1) it's Job Class System which they had been working through since the beginning of the series (& then again w/III & V) & 2) the SRPG format, which they had before only experimented with a little bit with the (non-NA) releases of Live-a-Live & Bahamut Lagoon. It had one of the truest translations ever (though plagued w/misspellings & the like) & a great, classic medieval storyline...But the main draw for FFT, of course, was in it's strategic-based gameplay. With the Job Classes, the skies were the limit to completely customize your party and try new ways to decimate the enemies. Easily one of the greatest heights of the series ends up coming from this series spin-off.
Riviera: The Promised Land - Easily my favorite handheld RPG (and just plain great game any way ya slice it). With a neat storyline, great characters, multiple endings, a weapons system borrowed from the SaGa Frontier series (it's a good thing, honest!), & a very strange, remarkably innovative (remarkable in the sense that innovation of ANY kind isn't a big draw in the genre) gameplay which can add to replay value, Riviera remains one of my favorite RPGs.
The Return of Samus: Metroid II - Gotta show some love to one of the best action/adventure/platform series ever. And this Gameboy (that's right, the old clunky gray box) classic is my fav of the series. Samus (who gets her name in the friggin title for the first & ONLY time in the series) goes to the Metroid homeworld and ends up having to fight each Metroid evolution, which ends up being some of the most intense battles of the series. It's ridiculous that they haven't remade this like they did for Metroid in Zero Mission...oh well, Samus can Screw Attack me any day.
Capcom Vs SNK 2: Millionaire Fighting 2001/Mark of the Millennium 2001 (Japanese name WAY better) &
Marvel Vs Capcom 2: New Age of Heroes - I really can't choose between these two fighting games, even though their gameplay is quite dissimilar. SNK is much harder, though much more customizable, which is nice. Both of them have a TON of characters, though Marvel edges SNK out. Plus, though I love Mai, Terry, & the rest, I've gotta say that Marvel totally fills my dork quotient pretty well. TWO Wolverines?!? lol
Unreal Tournament - Though I loved Quake III Arena, UT will always be king FPS in my heart. Good gameplay, awesome engine, great weapons, nice multi-player features...this game was what made FPSs cool before a certain Microsoct series (which shall remain nameless) came along and those console FPS fanboys started yapping about what they really knew nothing about...Well, anyways, as for how I play UT...hmm...Flak Monkey? Perhaps...^_^
Raiden - I gotta give it up for this whole series of top-scrolling shooters which got me through many hours of sitting in a bowling alley arcade. Easily my fav of the shooter genre. Gotta love that pink laser! ^_^
Armored Core 3 - I haven't played 4 yet or anything, but basically the best is the newest of the series because each one is just a touch better than the last. Tons of mech customization w/fun, addictive gameplay & an okay 2-Player mode for the AC junkies who tweek their cores to battle their buds. Longtime fan of the series.
Tokyo Xtreme Racer 2 &
Gran Turismo 4 - Along w/the PS2 port Zero & the 3rd installment, I think TXR is one of my fav racing game series. Lots of (unlicensed) cars & tons of customization features. The races were a neat way of doing things as well, including fighting game-like life bars that went down determined by how far behind the opponent you are or by hitting something. On the complete opposite spectrum is the Gran Turismo series. Lifelike handling & customization along w/TONS of licensed vehicles from every major distributer (and even the minors!).