I bought/acquired systems in weird order. This seems like fun, so I'll see if I can remember, from last to first:
PS3 Slim: Batman:AA, Uncharted 1, MLB 09 The Show, and FF7 on PSN on the same day
DSi: Bought Rhythm Heaven on the same day (and screen protectors)
360: My little brother (who likes the chronic) sold me his 360 with COD4: MW, COD2, NHL 08, and Saints Row 1 last September for $100. I bought Madden 09 and Portal off XBL the next day.
Dreamcast (weird order): Found it in a pawn shop with a controller and VMU for $19. I've only bought Dreamcast games once and it was that night, but it certainly was a shopping spree. Using eBay, half.com, and Amazon, I bought Daytona USA, Soul Calibur, Jet Grind Radio, Sega Rally, Space Channel 5, Shenmue, Crazy Taxi, UFC, NBA 2k1, NFL 2k1, and NHL 2k2. I won't lie and say I've played them all, however the UFC disc I bought on ebay is scratched to high hell and the Dreamcast won't read it. Kinda bummed about that.
PS2 (like I said, weird order...I missed a lot of last gen): Bought FFXII, GTA San Andreas, God of War 1, Shadow of the Colossus, and Fire Pro Wrestling Returns on the same day.
DS Lite: Costco bundle came with Yoshi's Story. I bought Phantom Hourglass at Costco 2 months before I got the system (I knew it'd be a Christmas gift) and bought Mario Kart and New Super Mario Bros off eBay before actually having the DS in hand. I also searched every pawn shop I could to get Earthworm Jim and Yoshi's Island for the GBA.
Wii: Costco bundle came with Mario Party 8. The next day I bought Twilight Princess, Super Paper Mario, The Bigs, and a Classic Controller.
A decent, modest PC: Once I had something that wasn't a hand-me-down piece of junk, I scoured eBay for any You Don't Know Jack game I could find. I've got most of them now, but no one around to play them and they're not much of single player experience.
Gamecube: For some reason, I'm remembering Mario Kart: Double Dash being packed in, but I don't know if that's correct. Costco had Super Mario Sunshine and Soul Calibur 2 for $20 each, so I got those right after I got the Cube as a gift.
PS2 (my first one): A guy from my church who's wife was kinda on him about spending too much time on videogames sold me his PS2 with Final Fantasy X for $100. I'm sure the first thing I bought for it was either Madden or MVP/Triple Play Baseball.
N64: Found/bought at a pawn shop because WWF No Mercy had just been released and I needed my fix. They had WCW Revenge there, so I bought that as well. During this time, K-Mart had Perfect Dark for $10 and I'd constantly remind myself to go buy it, but I never did.
PS1: Bought Crash and (I think) Mortal Kombat Trilogy on the same day. I also remember getting Street Fighter Alpha around the same time.
Super NES: Along with Super Mario World packed in, I bought F-Zero on the same day.
Gameboy: Tetris packed in, with the first game I bought being a Bo Jackson football AND baseball game. It was horrible.
NES: I didn't get Super Mario Brothers with my system. I got Duck Hunt and Gyromite. I think the first game I bought for it was Zelda 1.
Atari 2600: Lord, who knows. The games I remember having for it but no memory of buying/receiving as a gift were Pitfall, Jungle Hunt, River Raid, Space Invaders, and Missle Command. I remember my mom giving me Ms. Pac Man (which was a great port) as a gift after lecturing me about something. I also remember running home/opening up games like Pac Man, ET, and Indiana Jones and being so excited, only to be sooooooo let down.