My son turned three at the end of March and is also getting better at independent play, but he almost always prefers to have myself or my husband play with him. He generally wants to play with me more than I'd like to (or have the imagination to), but I try to always remember what Gretchen Rubin from the Happiness Project says, "the days are long, the years are short." And so I try to play anyway.
I stay at home with him and he's not currently in any scheduled classes or preschool, so it's really up to me to keep him busy. So, here's our day:
He wakes about 7 a.m. and spends half an hour playing with my husband before he has to get ready for work
My husband leaves a little after 8 a.m. and I sit and drink coffee with him in the playroom, usually reading email on my phone while he plays independently. He's learned that it takes mom a while to get going in the a.m. Around 8:30 or 9 a.m. we break for breakfast (although I'm trying to move this earlier, in anticipation of him starting preschool soon). Then we usually play together until 10 or 11 a.m.. We play cars, trains, coloring, games, pretend anything, hide and seek, go to the playground or a walk to the river, practice using the bike, etc. If I have errands to run, we'll generally run out around 10:30 or 11 a.m. and come back by 12:30 for lunch and then nap. He naps from about 2 p.m. until 4 or 4:30 p.m. and then we play again until daddy comes home around 5:30 p.m. Then the two of them usually go outside to play while I cook dinner. We eat and then have bath around 7:15 p.m. and off to bed by 8 p.m.
Putting it down on paper it sounds very routine, and I suppose it is, but what he wants to play with varies every day, and with errands and trips to the playground or the library thrown in, the days seem to just fly by.