Games I like that you can be invisible in:
Geist - Game Cube
In Geist you're a scientist who's been turned into a ghost in a crazy experiment (if I remember correctly, it's been a while since I played). You can fly through doors, possess items, and scare people witless. Very good play.
...
Um, there was one of the Marvel games, but I can't remember the title of 'em (because I know they made more than one), but the Invisible Woman is a playable character. A friend and I were playing, and were stuck at this one part where a bunch of baddies were guarding a very essential piece of equipment that needed to be shut off. A friend and I had our butts handed to us once or twice, so I said, "Screw it!" and switched over to Sue Storm. When we reached the objective I simply cloaked, snuck past them, and simply flipped the power switch. I felt rather empowered and clever, but my friend seemed to have mixed feelings about it.
Team Fortress 2
Everyone's favorite class is Spy, hands down.
Othr games I know of are Tenchu, Wrath of Heaven, and The Hobbit.
But in none of these games are you permanently invisible :*(.