Growing up playing early console games, games like Super Mario 64 ensured my enjoyment of console gaming for years to come.
In more recent terms, when I first got my Xbox 360, I had never played any online mulitplayer titles before. The first game I bought was Call of Duty 2, and after playing with my friends for hours on end, my views of online play were expanded on much broader terms.
EDIT: Also when I was younger, I remember Paper Mario being the first game that completely stumped me (mostly due to the fact that I was born after the popularity of incredibly hard classic games) thanks to that pyramid level with a confusing layout. Yeah, not really so much of a positive imapct as the other two were.