My first thought was that the child might have
Prader-Willi Syndrome, which is one of those rare and hopeless genetic disorders that it is fun to pretend having, but devastating in real life.
Basically the sufferers don't have the right "switch" in their brain to tell them they have eaten and it gives them an insatiable appetite. They are constantly hungry, with all the misery that entails. And there is almost nothing to be done for them - they die young from obesity related illnesses. Coupled with learning disabilities and emotional immaturity it is very hard to treat apart from a draconian control on food intake.
Anyway, reading on I realised that the child's growth was also an issue, which later posters addressed.
Thanks for the info!