I babysit on a regular basis for a couple who has 3 kids. A 5 yo who listens when he wants to, a 3 yo who has some sort of very mild autism, and a 2 yo who has her moments. She's very stubborn. When she's happy she's a bundle of joy, when she's not, look out for screaming bloody murder (like when mommy just walked out the door). Anyhoo, i was watching them with my friend once (I'm almost 15, friend is 15. i didn't feel comfortable watching them by myself since it was all day), and we were playing with a sheet (of all things) and one of them would get in the middle, we'd pick up the sides and swing them back and forth. Now every stinkin' time I watch them, it's "swing, swing, swing" and since it's just me, I can't do it (since you need 2 older people), and the younger 2 scream. I also have another problem when I got them dressed up in swimsuits one time, and we laid down towels and played with water in the kitchen floor. now every time it's "water, water". hold on..... typing.....
I'd find some other less messy fun, like getting bowls of rice and beans and hiding little toys in there. Get a rope out and play snake hunters (you drag the rope around and they get to try to catch it, jump on it, whatever) Get the sheet out and play parachute, with the older kids (he'll listen if you say I've got a fun game!) Let them take turns going underneath. Put different stuff on top and see how it bounces. Put on loud music and do funny dances (which is good because when you're tired and sit down, they'll keep dancing!) We also do blue's clues, where we put blue spots on different stuff and they find them. Tell them they can play water in the tub, and put all three of them in there with some toys (added bonus, mom's love coming home to a set of clean kids and soaking for an hour in soapy water, you don't even have to scrub!)
They just like the other games because it is really intense fun. A movie isn't ever going to replace that because the kids want to do fun stuff. Messy stuff take outside (finger paint can keep my kids for hours, and if it's outside I just hose them off and leave everything out there)
Good luck, you sound like a babysitter they really love to have around.
You've got your job cut out for you! When you arrive, you need to have a plan. As soon as the children start w/ the swing or water request, you must have a diversion. Without hesitation, tell them NO water or swing today. Get up and go in the other room, come back w/ a new idea. Now, we're going to color, read, watch a movie, etc. Keep this up and they will learn that they can not bully you into doing what they want. Don't give in.