Hmm...
I can't argue with Devil May Cry 1, but I would leave 2 and 3 alone. Also, God of War is a decent choice, as is Guitar Hero.
For the X-Box, if you can find it, Pranzer Dragoon. If not, look out for Elder Scrolls if your an RPG fan, Mortal Kombat, Armageddon or Shaolin Monks for some pick up and play action, Black is a decent shooter, but if you can find Halo and play that to death.
Gamecube, Twilight Princess and Mario Party are musts, and I would also suggest Killer 7 and Beyond Good and Evil.
For the 360, a nice action game would be Dead Rising, a nice shooter would be Gears of War, and a nice Sim game would be Viva Pinata. Oh yeah, and Saints Row just for some fun