Hi Ho Cherry Oh!
Hungry Hungry Hippos
Trouble
Cootie
That's what I could come up with spur of the moment, I used to be a nanny. The first one and last are the best, but both have a lot of little pieces. Other games that are simple and fun for little kids: Button, button, why's got the button? Paper Rock Scissors (suprising, but they seem to love it. However the 2 yr old we taught always did a thumbs up for the first 100 times she played

it was cute!) And that "Fish" game comes to mind where you each have a fishing pole that has a magnet on the end and you catch fish that open and close their mouths with a magnet in the middle.