Obviously, soccer is right out--the ball is just way too big. Similarly, Monopoly takes way too long for someone with such a short attention span.
You may be asking yourself, then, exactly how DOES one play with an infant? Considering the limitations of the baby and similarly the over saturation of the market with useless items (see consumerism convenience and consomething else) it's hard to know what to buy, what to do with it, and when it's appropriate.
The simplest way to "play" with your baby is to touch him. Making physical contact is an important lesson your baby needs to learn. Skin on skin is essential for your baby to bond with you, and c'mon, who doesn't love a good massage every now and again?
Another easy way to stimulate and amuse baby is talking. Some people (yeah, that's me referencing myself there) feel a little silly at first, talking to a baby. The simplest way is to start small. I found "hi there, I'm your mom" to be a good lead-in. From there you can talk about your day, your sordid past (it's ok, chances are pretty slim that anything you say will be retained) or just go with a running commentary on your actions. I always swore I wouldn't be that mom babbling day in and day out in baby talk, but guess who I turned into?
I even made up a whole series of "------'s a Baby" songs, revolving around the parts of the body, and whatever E was doing at the time.
Tummy Time is super important, since the introduction of the "back is best" campaign and "back to sleep" positioning suggestions have shown a decrease in advanced learning of large head and neck motor skills.. Tummy time is when you put the baby facedown and SUPERVISE CAREFULLY while he learns to do things like roll over and pick his head up. They make some specific Tummy Time Mats, but anything clean that can be drooled on is ok. Since it's not very engaging to just be placed on your tummy, try to keep your baby's attention during tummy time. get down on the ground, talk to him, play with toys, or just make faces.
Teaching motor skills can be fun! We used to do bicycle legs to release gas and promote hip flexibility, and now we do it because it's fun. He helps me more every time. I figure by the time he has a pedal bike he'll be an old pro.
E also has a baby gym, a pretty simple one. Every day he lays on his back, staring up at the toys dangling in front of his eyes, flailing wildly until he makes contact. Every day the flailing gets less wild, more controlled, more calculated. Last night he reached out and grabbed one of the support arms and just held onto it for a good three minutes.
Most dangling things will be very interesting to babies. Mobiles, gym toys, keys, hair, earrings... some of these things are good. Some are less good. to take the focus off the less good, make sure the more good are brighter, shinier, and have more contrasting colors. Also keep the really not good out of reach. My mother has a torn earlobe because of not having earrings far enough away from my hand.
Remember the first time you ever felt a bunny, or a chinchilla? Every texture is like that for babies, so find some interesting textures on large items (no choking hazard here), sanitize them, and introduce baby to some fabulous new feels.
Playtime isn't just about amusing your baby, it's about learning to use this fabulous new body they've been given, learning to interact with the world, and learning that naptime can be fabulous after a long day of activity.
And trust me, naptime is FABULOUS.