From the top, my suggestions.
SNES
-Super Mario All-Stars, for being many great classics in one cart
-Legend of Zelda: A Link to the Past
-Super Metroid
-Super Punch-Out!!, for being so cheesy and yet so fun
-Final Fantasy II (or IV Hard-type, if you emulate)
-Final Fantasy III (a.k.a. VI)
-Chrono Trigger
-Tales of Phantasia, although I prefer the battle control scheme of the Playstation/GBA remake, but the DeJap translated ROM is more true to the original version
-Seiken Densetsu 3, which I consider to be one of the top three RPGs on the system - yet it never made it out of Japan officially ;_;
-Super Mario RPG
N64
-this system in general, for making multiplayer gaming easier and more fun for me and my friends. Wish there were some decent RPGs for the system here in the States.
-Super Smash Bros., the original. Easy to learn, yet well-balanced. Playing against human opponents is the best, naturally.
-Starfox 64
-Mario Kart 64
-The Legend of Zelda: Ocarina of Time. I shouldn't have to explain this one; it's one of the great masterpieces of its time. I didn't care all that much for Majora's Mask, though.
PSX
-Castlevania: Symphony of the Night
-Twisted Metal II (Aside from TM:Black and the PSP game, all TM games after this one sucked hard)
-Final Fantasy VII. Yes, I said it. I still like it, despite its faults (and the way Squenix seems to want to run it straight into the ^&(*ing ground...)
-Tales of Destiny, for introducing me to a better version of Phantasia's battle system
-Tales of Eternia, though I wish there was an option to not use the English language dub track
-Breath of Fire III. Best game in the series, IMO.
-Final Fantasy Tactics, though I never did finish it.
PS2
-Disgaea, Makai Kingdom, Disgaea 2, and to a lesser extent, La Pucelle. All great games, if this genre suits your fancy.
-God of War. I still love this game. Wanton violence, murder, callous disregard for life, blood and gore, and a sexual threesome (uncensored topless women included!)... and that's all just in the first level!
-Dynasty Warriors 3 and 5. Expansions optional. Flashy hack-and-slash fun, with a historical-fiction twist.
-Samurai Warriors (expansion NOT optional; it's too awesome). Same, plus greater difficulty, more interesting mission-oriented gameplay, and more.
-Guilty Gear XX Slash. Nevermind the suggestive title; it's a freaking awesome game. Revitalized the 2D tournament-fighter genre, has a great deal of depth to master, possesses a colorful and bizarre cast the likes of which hadn't been seen since JoJo's Bizarre Adventure (and a world just as colorful), and has a hard-kicking soundtrack largely inspired by the great rock and metal legends. Full of Win.
-Soul Calibur II. This game has a few flaws, but they are vastly outweighed by the intuitive, simple, yet very deep gameplay engine and smooth graphics. Just avoid the English-language dub track; some of that stuff is painful.
Also, Sophitia and Ivy make for some incredible hawtness. ^_^
(Unfortunately, despite the larger cast and Create-a-Character option, as well as a good English-language track (Crispin Freeman makes a damn good Siegfried), Soul Calibur III is plagued by problems - e.g. lame CaW movelists, cheap AI, lack of selectable difficulty setting, and the infamous Corrupt Save-data Bug - that just ruin it. Fail.)
-Guitar Hero. You'll never be cool playing it, but the songlist is awesome, the gameplay is fun and addictive, and it's easy to get into it.
(The less-than-spectacular tracklist and screwed difficulty progression of Guitar Hero II make me unable to recommend it, but it's worth a play at least.)
-Lego Star Wars II. It may have a cheesy premise and all, but it's fun and addictive, and full of unlockables.
Gamecube
-Super Smash Bros. Melee, for all the same reasons as the original, and then some.
-Tales of Symphonia
-LoZ: Twilight Princess. This game is great for all the same reasons as Ocarina of Time, looks better, has deeper controls... all in all, it's a fantastic farewell to the poor GC.
-Naruto: Gekitou Ninja Taisen 3 (import). You need a loader or modchip to play it on non-Japanese 'Cubes, but it's worth it for the fun of the multiplayer. I played it at an anime convention last year, and wished I could've played more.
(FYI, GH, it's set before the timeskip, the Sannin are all playable, and there are plenty of extras that don't suck.)
-too many other unsung heroes of gaming. The 'Cube just hit bad luck on the industry. Isn't it sad, Gamecube? ;_;
PC
-Unreal Tournament: 1999 Game of the Year edition. There's still a lot of user-created content for it, and LAN gaming drives its goodness.
-Starcraft/Brood War. Because it's still a great game. Still awaiting SCII.
-Dawn of War, and expansions. Also a great RTS.
-Melty Blood, MB:ReAct, and all updates/patches therein. It's just fun to play, when I can get to my friend's house (curse my inferior system specs!).
-All other Type-Moon games. Damn, but I can get into Nasu's works!