When our son was that age, we would sometimes:
* play Texas Hold'Em poker
Tara
This approach can lead you into trouble. In addition to chess, walks, and many of the other things you mentioned, we played poker with our 5 yo kids. After a stretch of poker playing, we happened to have a doctor's appointment for an annual exam. As the nice pediatrician tried to talk directly to my son, she asked him about school, his bike, interests, etc. When she asked him about what games he liked, he replied brightly and with a huge smile, "I like to play poker for money!" I turned bright red and felt really embarrassed, but it was true -- we did play poker for coins, and he did enjoy it!
J