Home console: The original PlayStation. Our household opted for the Nintendo 64, and while I was satisfied with it, there were a number of PS1 exclusives that struck my interest. There was a point where I could have got one as a birthday gift, but by that time, the PS2's launch was around the corner and I decided to wait for that and its backwards compatibility. I eventually acquired a original PS1 for $4.99 at a Goodwill, and last year, I bought a PSOne with the official LCD screen for $35 at the Portland Retro Gaming Expo.
Portable: Game Boy. My mom wouldn't buy me one because she didn't like the screen. She did get me a Game Gear, though, and I was pretty satisfied with that. To this day, I still don't own an original Game Boy (although I do have two Game Boy Colors - which were also $4.99 Goodwill finds), and didn't start collecting original GB games until I acquired my first Game Boy Advance in 2004.