The Pokemon and Zelda series, particularly Yellow and Silver for the former (Fun fact, I didn't have a Game Boy when I bought Yellow, so I played with Stadium on N64), and Ocarina of Time and The Wind Waker for the later. Star Fox Adventures, Dragon Quest III, Paper Mario, and Yu-Gi-Oh! (You said games, not specifically video games) where highly influential and enjoyable to me as a kid, as well.