Not Napping Three Year Old

Send him to preschool.
At this age, they often need it.
They developmentally need more socialization and being with their own peers, and learning other things and about routine/behavior/listening to a teacher.
When I had my 2nd child, my daughter went to Preschool. Part time. It was not to get rid of her once I had a baby underfoot, but... SHE needed it. She was even asking to go. And she LOVED it. I was HER time. And she gained a lot of positives in preschool. And then it also prepped her for Kindergarten.

And, when either of my kids were napping, they were quiet. They understood, to WHISPER when one was napping. And every afternoon, when they were younger, was "quiet time" and for Mommy too. They understood and did it. It was done anywhere in the house, and as long as it was quiet things. They didn't battle about it. It was the "routine."
But often, BOTH my kids, napped at the same... time.
And even at that young age, children learn and understand ROUTINES of the household. ie: Mommy needs to cook/clean, they have a quiet time, then there is meal time/play time etc. EACH day, there are certain times... for things. And they understand, that they can be self-reliant or learn, to be independent... ie: while Mommy is in the kitchen cooking or cleaning. It is, cooperation. When my kids were younger, I would tell them "Mommy is doing her routine now.... cleaning the kitchen and preparing lunch..." and they understood. I used, the SAME word cues, with them when they were younger. So that, it became routine. And they are now 6 and 10, and all I have to say is "Mommy has to do her routine..." or "quiet time now...." and they understand. Fully. What that means.
They need to be taught, by words and examples and by doing it everyday. So that, they know the everyday, rhythms of the house and of Mommy and of them.

I tell my almost 4 year old (next month) he doesn't have to sleep but he has to stay quiet in bed for an hour, his "quiet time." He can play with quiet toys (stuffed animals or a couple of small action figures) and read books. No making noise, and I don't sit with him.

I've explained to him his body needs rest even if he isn't tired, and that he needs to be respectful of others resting or napping at that time. If he makes noise or gets out of his bed or his room, he has a consequence like not watching a fave TV show that afternoon or not playing on the Kindle or going to bed earlier.

At almost 4 he's old enough to behave or have a consequence, it's my time to get things done, too.

I sent my kids to afternoon nursery school three afternoons a week at that age.

When my younger son was still napping I often sent my older son in to the back yard to play. I could crack the window to listen for him, and peek out on him regularly while I worked.

One hour of tv a day won't hurt, and it will be a quiet time for him to relax and recharge.

One thing that's always worked for keeping my children occupied for some length of time is water. Fill up a dish tub with water, and give him cups, spoons, bowls, water toys, etc. Put a towel on the ground and let him go to town...someplace away from the others, like in the kitchen. The mess is surprisingly easy to clean up.

I know you said art, but I'm not sure what you've tried. My little ones have enjoyed markers, stamps, stickers, clay, and a random assortment of other craft supplies.

Video games? I know some people might not approve of that suggestion, but if I want my toddler quiet for an hour because I really need to get something done, etc, I stick the DS or the Leapster in her hands and she's content. It's not something I do very much, so I certainly don't feel guilty about it!

Have him help. He can help set the table, sort laundry, dust with a rag, dry dishes, etc.

Part of this is him needing to learn to entertain himself. You have to sort of refuse to be his entertainment! ;)

I know it will slow you down, but consider having him "help" you with your own chores. He can fold laundry (or pretend to) and get it into a basket while you're next to him folding too. Give him a bowl and a wooden spoon and let him stir something for dinner or tear up lettuce with his very clean hands to make a salad. If you have to measure something with measuring cups, fill them but let him dump them into the mixing bowl. If you treat these things as chores and bores he will think of them that way too, but if you make it all fun stuff that ONLY he gets to do with you because he is the big boy - he will love it. You won't get as much done, true, but you will be using this time wisely, as an opportunity to be alone with him -- something I bet you don't get when the other kids are awake.

So you don't have to be "sitting with him contstantly." Instead, be up and about with him and keep the little tasks moving fast, so he doesn't get bored. On weekends prep whatever you can and freeze things for weekday dinners (chopped veggies that take a long time to chop, etc.) so you aren't doing that when he's up and with you.

Do try to see this as your opportunity for one-on-one time plus a chance to teach him that these things can be fun to do.

I know this is NOT what you asked, but is there any opportunity with his age and availability to be in a part-time early child care learning center or MDO program to help prepare him for the classroom environment and give you two days with napping little ones and no big one to entertain?

It could be too early at three, but at four it could be a real opportnity for you.

Other than that, I agree with the ideas of water and the quiet time even with no sleep:)
good luck.

Make that his quiet time. A DVD or TV is not the end of the world. It will save your sanity or like A L said have him help you do things. I bet he would love that and he is not too young to start helping out with little things.

Well, if you can't beat em, join em.. by that I mean.. if he won't nap and wants to spend time with you.. then let him begin to help you.. if you are folding clothes.. give a few towels to work on. if you are prepping dinner, let him do something (maybe hand you things) Clearly, he can't do much at this age.. but this may a situation where you just have to change up your routine..

Also, this time could be looked upon as yours and his special time together.. kids grow up quickly... in a way, this could be a blessing in disguise... so often you hear parents say, gosh, I wish I had more time with my kids.. OR I'd give anything to stay home with my kids... maybe try and see this time as quality time that has been given to you (thanks to your son :) ...

good luck

In child care when we have school aged kids in the center while younger kids are napping we start a movie and make some popcorn. They sit for at least an hour an a half quietly and don't bother anyone.

You might also want to do some one on one time with him during this time. He's bored with no one to play with. Research PCIT and see if there are any hints in that therapy that might be a good idea for you guys to play with. I loved playing with play doh or made craft stuff for my special time with the kids, hubby did lego's, blocks, and some other stuff.

It's a good time to spend with him uninterrupted.

Let him lay on the couch and watch part of a movie. That's the ONLY way I can get my soon-to-be 5 yr old to have "quiet time". He's good with books/quiet toys/puzzles, but only at the dining room table where I can see what he's up to. Otherwise, the noise level goes up pretty quickly, which defeats the whole purpose.

My guy isn't much of a t.v. kid, but he likes to lay down and snuggle with a bear for a movie on the weekends.