what is normal??? each kid is different. I had at least two that quit taking naps at a younger age than I would have hoped for. but oh well.. that is just the way they are made.
if he is old enough to tell you he is not sleepy, then he is old enough to understand some simple instruction. And this is what I did with my no-nappers: I told them they must have some quiet time. They did not have to sleep... but they must play quietly, look at books or another quiet activity in their room for a set period of time. if you want a break to get some quiet time yourself.. then maybe save that period of time for him to watch something on TV.. if he will sit through a show. Invest in some good videos he likes. Another idea is to see if there is a neighborhood pre-teen (11 or 12 years old) that would be willing to take him out to a nearby park or on a walk...or play with him in the yard. This age group loves kids that age and would probably do it for free... it will give them some experience for future babysitting jobs.. they can take a red cross babysitting course at 11 so they might already have experience in dealing with kids. My 12 year old does some paid babysitting but loves kids so she does some to help people out if she knows they could use the help.
another thing to maybe put things in perspective..at least he sleeps through the night for the period of time that he does. I had one that quit taking naps before she was even 2.. and she still got up a couple times a night.
oh yeah... and maybe see about putting him in a preschool. It would give you a bit of a break and help him "spend" more of his energy a few days a week. Look into your local Head Start program and see if you would qualify to send him there (it is typically income based). You could also have him evaluated through the early learning intervention program at your local school district, if they qualify him for services (he would then have an IEP) - then he may still go to Head Start for free even if you don't qualify based on income. It is a terrific program for kids, my 4 year old goes there - it is worth it if you can pay for it. I wish I would have gotten some of my kids in it when they were that age.. at least the "no-nappers" would have benefitted from it greatly. I was told here.. that the public schools know which kids come from the Head Start program when they enter kindergarten because the program is so good for these kids. I am very anxious to see how my son does when he enters kindergarten this fall.
best wishes!!