Some great contributions in here. My absolute must haves are pretty small, as I'm generally not easy to please.
Wii
Sin & Punishment Star Successor - Amazing 3D shooter, and a miracle that it even got made. More like Starfox Successor m I rite?
Zack & Wiki - As an early Wii game, the motion controls at times feel clunky, but it's an absolutely original, fun, entertaining title.
Tatsunoko vs. Capcom - The best fighting game on the system, and the most fun and approachable Capcom fighting game since Rival Schools.
Little Kings Story - Beautiful, charming, fun game, and an overlooked RPG. The Yuri Norstein inspired artstyle and classical music-derived soundtrack are straight after my heart.
Wario Ware Smooth Moves - Came out to mixed reviews, but it's the definitive Wario Ware console experience, and one of the best games in terms of motion control utilization. The best multiplayer game for my money on the system.
There's lots of other great stuff that's certainly must play material if you don't have them on other platforms, such as Resident Evil 4, Metroid Prime Trilogy [1], Twilight Princess, NPC Pikmin 1/2, Xenoblade (coming out for N3DS), and Smash Bros Brawl (if you don't have a Wii U or 3DS for the superior successor).
DS
Pokémon Black/White - The best and freshest Pokémon game since R/B.
Castlevania Order Of Ecclesia - For now, the metroidvanias final form. It's more action oriented than the prior few entries, and has some of the best art of any game ever.
Zelda Spirit Tracks - Tremendously better than Phantom Hourglass. It has perhaps the best set of items in a Zelda game in terms of utilization and distinctiveness. Great graphics, charming plot, fun mechanics, and the perfect amount of content for a handheld Zelda.
Advance Wars Dual Strike - Not as well balanced as other titles in the series perhaps, but it has a tremendous amount of content and never lets up on entertainment.
Rhythm Heaven - The best rhythm game Nintendo has released outside of Japan. Enough said.
Tingle's Rosy Rupeeland - Completely fucking ridiculous, but absolutely hilarious and irresistible. Disclaimer: you essentially have to play it with a pricing FAQ, but it's a very fun and completely unique game. Only released in JP and PAL territories.
I realize that OP has many of these, but thought I would type out my complete must-have list.