Man, this is a tough list.
My favorite classic games:
1. Final Fantasy VII
What can I say? It was the first FF game to enter 3D and it blew my freaking mind. The story is so crisp, characters awesome, and nothing, even to this day, beats the Materia magic system when it comes to Japanese themed RPG games. It is a masterpiece of classic RPG gaming, with every single tiny little thing done to please whomever is playing it.
2. Fallout 1 and 2
Fallout 3 and New Vegas are pretty great, but for those who haven't played the first two, are really, really missing out. The second one in paticular is quite amazing. So much to do, so many characters/interesting things. It also has a lot more humor and crazy things. I love the first two more than I love 3 and New Vegas. If you haven't played them, go buy them on steam for 10 bucks or whatever it is, they are serioulsy great games.
An Amazing game. Seriously. This game changed first person shooters forever, more so than Halo ever wishes it did, and to this day the Half Life series is by far the best FPS series ever made. For anyone interested, they (community) are remaking this game with whole new graphics, and it's free so long as you have the original Half Life.
Link to remake is here:
4. The Secret of Monkey Island 1 and 2
This game is pretty dang fun. No combat to speak of, minus the whole "pirate fighting", which is hilarious and entertaining. It's a pure, classic adventure game. Items to be found, puzzles to be solved, and (goofy and amusing) story to unfold. Both games are so great.
5. Ocarina of Time
It is a good time for this thread, as I just picked up the remake for this game on the 3DS. The game's core is literally as good as any new Zelda you will find. The combat is great, the quests are fun, the dungeons are tough (if you didn't already know them font to back like myself, that is), and it's a long 30-40 hour experience to get everything. It's one of the best games ever made, and rightfully so.