Playing a game with a newborn is a little tricky, but being a game-loving family, we try our best. Here is something my husband came up with for M when she was a baby that we do now with Baby G.
When the baby is sitting in her bouncy chair, my husband tosses a baby blanket over her head. She, of course, has a pretty constant flailing of arms, and soon the blanket falls down from her face. We exclaim peek-a-boo, and usually earn a big smile- sometimes giggles. I don't imagine there's any connection at all right now in Baby G's mind between her arm flailing and the peek-a-boo, but I remember that with M, it really seemed that over time she grasped that there was a cause and effect involved.