Here's what I do : I stock up on store-brand pretzels, animal crackers, rice cakes/crackers when they're on sale. The kids get what's planned on my menu for meals and snacks, but if they're still hungry and have eaten everything on their plate (and anything else left over), I let them eat from my stash. However, I only feed them at meal & snack times. If they get up from the table, they're done for that meal-time, and they have to wait for the next snack. I serve breakfast, lunch and 3 snacks, so there's no way I'm letting them snack in between! If they're really hungry, they'll eat at meal/snack times. If not, I figure they're just looking for attention
Still, if a child is actually hungry and is willing to eat whatever we offer him, I don't think we should be denying him more ...