I'd say at least 10 oz of water daily shouldn't hurt. As long as she is getting most of her nutrition through breastmilk (or formula). Water only becomes dangerous when it replaces the nutritional feeds. never let her fill up on water before a feeding, but cups between feedings or with food is okay. As long as she keeps nursing enough to keep up your milk supply, she'll be fine, just remember with breastfeeding, it's all suply & demand. Whatever she demands, your breasts will supply, so the more water she drinks, the less she suckles & the less milk you'll make. Tiny stomachs need frequent refills. But, there is a endocrine disorder called diabetes insipidus, which causes a constant thirst. It is rare, but if it seems like she constantly wants to drink, no matter how much she gets, maybe wise to have a pediratrician check her out. It really doesn;t sound nerly that serious to me, but hard to tell in an email, so jut thought I'd mention "worst case scenario" possibility.