With the Down's and ADHD, this may be more of a behavioral issue rather than a hunger issue. Does your daughter communicate to you that she's hungry, or just that we wants to eat? Does she eat large portions? Does she only do this when there is a trigger (ie...you in the kitchen, or the oic of food)?...or does she do it at random times, independent of a verbal or visual cue?
Have you tried telling her NO? If so, how does she react?
I would definitely take her to the pediatrician, rule out any kind of medical condition (a parasitic infection would be rare if you're in a developed part of the world). You need to determine whether this is actual hunger or behavioral. I have a sneaking suspicion it is the latter. If that is the case, you're going to have to do some behavioral modification...set limitations, adhere to strict meal and snack times....explain to her that she is getting enough to eat and should not need to eat that frequently, and explain in simple terms that there are ramifications for overeating.
As a new nurse, I worked with a special group of children with a rare condition called Prader Willi syndrome. I'm not at all suggesting your daughter has this...but so much of what you said reminds me of those children. Look it up...if there are some similarities, it may point torward the behavioral issue.
Very best to you...I'm anxious to hear what the doc says.
Children generally eat more then adults do, although by what you are mentioning here, she does seem to have an abnormally heavy appetite,
My question to you is, does she have a lot of digestive problems? Does she get diarrhea very frequently? How is her weight at this age? Is she gaining weight? Or is she losing weight despite eating a lot.
A parasite infestation of the GI tract can lead to increased appetite and simultaneously no weight gain. So discuss this with your doctor, they may give her a course of metronidazole or diloxanide furoate.
There is a condition called pica where the appetite is increased abnormally and children are even spotted eating mud and other non edible stuff. So monitor her for eating things that are not edible.
Among her drugs, none of the drugs she is taking causes increased appetite very frequently. Synthroid being a thyroid hormone can increase her metabolic rate and that could lead to increased appetite, but it's just me trying to connect the dots here, it is not documented as a frequent cause of increased appetite.
Clonazepam can cause increased salivation but whether that leads to an increased appetite is debatable. Again it is not a documented cause of increased appetite.
It is most important to relax and to try to take a problem solving approach towards your fears for your child, hope this helps.