tremere110: Saints Row 4 because it's absolutely hilarious.
In this vein - check out Destroy All Humans!
I played the first one on Xbox and the game was fantastic. Open world (ish), with a huge array of powers and weapons, all of which controlled smoothly and were both a ton of fun AND funny.
Your little grey alien could do everything from read minds to camouflage himself as any NPC to jump into his saucer and level city blocks. It even featured a "force lift" type power that worked perfectly with the two sticks of the Xbox controller. Super fun to pick people up with your mind, spin them around in circles twenty feet above the ground then fling them LOL. There was a sequel or two released that for some reason I never played. Not sure why as I loved the first one.
*edit my post to add: Hey good to see Destroy All Humans! getting a lot of love here! I just scrolled up to read a few of the responses and saw it hadn't been forgotten!
Exoanthrope: I can't really think of many alien invasion games myself, except for
X-Com and
Destroy All Humans! if that counts.
zevri: I had a blast with X-Com Enemy Unknown. I also liked the old Destroy All Human's series. The first was fun for the tactical combat, which could drive you up the wall if you got unlucky. The second was fun for good old go on a crazy rampage.