Well, I own a tonne of consoles, and became a collector in the earlier part of this decade, but growing up there was only one game that made me buy a console I didn't originally want, as I was a hardcore Segabot.
This game has made me the gamer I am today, and opened my eyes to the glory of multi-console ownership...
that title is...
Mario Kart 64
Once I played it, I HAD to own a 64. Up until then, I had only owned Sega consoles, and had a Saturn at the time. I sold my Saturn (bought another one later on) and bought a 64. It made me realise how great multiplayer gaming could be, and made me realise owning octher consoles aside from just one meant I would never again miss out on amazing titles on other platforms.