A butt child is a a weak minded follower of another individual normally a partner/boss at work. He/she has no free will or independent thought but is completely infatuated with their icon and follows them around unthinkingly. They agree with everything their idol says without hesitation like a child.
Often a weaker member of a particular nauseating couple who dotes upon their partner unaware of what this looks like to others.
Annoying acolyte who imitates their masters actions.
Did you see the way that douche followed her round all night, jeez what a Butt child.
At work Mike agrees with everything Rob says. Mike is Rob's Butt child.