Can I just count entire series as one game? Well, better to seek forgiveness than permission.
5) Pokemon Spin-offs. Enough's been said about the main saga, but few acknowledge the highly underrated spin offs. Not all of them were good, but I highly recommend Pokemon Snap, Pokemon Ranger, and Pokemon Colosseum/XD (darker storyline and less grinding)
4) "Hotel Dusk" and its sequel "Last Window". A point-and-click interactive story. I understand you might not be into that but I am. Both games feature very engaging stories at that
3) Animal Crossing (Gamecube only). Yep, this would appear on my list given my namesake. I know I said I'd be recommending series but for this I can only recommend the Gamecube version. It's like The Sims, only you play as a character not a god, so it's more of a challenge. As I said I don't recommend the DS or Wii versions, I felt the new features made it less fun.
2) Thief. Ever wanted a first-person game where you could actually enjoy the scenery without the blinders of a constant hail of gunfire? I'd call this the only true stealth game becuase unless you're playing easy, you cannot fight your way to victory. Characters are engaging, missions are fun and challenging. Oh and there's a forth installment being announced.
1) Paper Mario. I have a soft spot in my heart for well-written dialogue, and this series far-and-away delivers. It's a laid-back RPG style with plenty of secrets to find and lots of replay value.