I just wanted to hop in here and say that although rare, it is possible that children do not know when they are full. There are lots of reasons for it, too.

I recently provided care temporarily for a boy who is currently in foster care. He had been starved as a child, and as a result, he will eat himself sick if allowed to. Sweet kid, sad story.

Most of the time, though, it is because of the situations discussed above. Mom and Dad assume fussiness is hunger and practically force feed the kid. Or they just shove food at him all day long. He never gets to actually feel hungry, or full.

There are rare disorders in which the brain is always telling you that you are hungry, even when you are not. That doesn't sound like the problem though.
Given that he is one and not even crawling/scooting, I think that this is part of a larger developmental delay?