In order of acquisition:
NES: I never owned that famous Mario/Duck Hunt cartridge until much much later, so when I got my NES, I bought Popeye (it was cheap, sue me) and Super Mario Bros 2. The latter was pretty much the reason I bought the console, and MAN did it not disappoint. Good times.
Gameboy: Tetris, since it came with the package.
SNES: Super Mario World. Obviously. Possibly the best launch game ever.
N64: Super Mario 64. Again, pretty obvious. Was really into it for the longest time, but in hindsight it's probably one of my least favorite Mario games.
Playstation: Resident Evil. The very reason I bought a PSX. I rented the machine and this game once before, just to see what it was about, and it was AWESOME. The series has gone through a lot, but the first game is still one of my favorites.
Genesis: I was a bit of a late-comer to this console, having been a bit of a Nintendo fanboy until the N64 turned out to just be kind of ass. I had played Sonic games before though, and this is the main reason I decided to pick up a Genesis on the cheap in Japan. I think I bought it together with the 4 main Sonic games. Those games are still fantastic to this day.
GBA: Castlevania: Circle of the Moon and Mario Kart. Ended up hating CotM (incredibly frustrating and way too dark to see) but the GBA Mario Kart is my second favorite in the series after the SNES original.
Sega Dreamcast: Marvel vs. Capcom 2. The main reason I bought the machine. Amazing game, but I totally can't get into it now anymore.
Gamecube: Resident Evil remake. To this day, probably the best remake I've ever played. I could do without those goddamn Crimson Heads though.
PS2: Street Fighter EX 3. The PS2 had a sucky launch line-up in Japan. This one was the least bad. Sold it off not too long after though, because, yeah, it's still bad.
Sega Master System: Picked it up for 2 Euros (!!) at a flea market, but it didn't include any games. Still haven't bought any either, haha.
Nintendo DS (Lite): New Super Mario Bros. It was okay, but not nearly as good as any of the 2D Marios before it.
Wii: Legend of Zelda: Twilight Princess. I seem to recall liking my first playthrough, but couldn't get myself to go through it again. Never touched it anymore since, and have no real desire to.
Xbox 360: Blue Dragon. Got lured in by the names of Sakaguchi and Uematsu, and although Uematsu definitely did not disappoint, Sakaguchi was kinda running on an uninspired auto-mode. Not bad, but meh.
PS3: MGS4. Hands down my least favorite MGS to play, but man were the cutscenes ever awesome.
PSP: Dissidia. I actually won a PSP, but I won it right around Dissidia came out, which I was really interested in, so that worked out excellently. Dissidia is probably my favorite fighting game series ever, by the way. So awesome.
3DS: Professor Layton and the Mask of Miracles. It was the only launch title that seemed worth owning, and it was another entertaining puzzle romp, certainly. It also serves well as a title to show off the system's capabilities since it looks pretty and the 3D effect makes it look like a diorama type thingy.