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When a rescue rat of mine was pregnant and gave birth, I could handle her and the babies from day one and she didn't mind at all. My housemate kept three of those babies, all girls, and they are fantastic pets, and have never bitten. Heck, even their 'tasting' doesn't hurt, they don't use any pressure on their teeth. They'd rather lick you or steal nummies from your plate than bite! So most rats are not biters.
Anyway, OP, I definitely suggest rats. I have experience with both rats and chinchillas and I'll be honest, chinchillas are soft and cute, but awful pets. They are more of a look but don't touch. They stress out easily, have a very specialized diet, and just don't like being held or touched really. Even ones raised from infancy are aloof. Rats however, love their people. At night my housemate would release the rats (raised by us from day 1)when everyone else was in bed, and they would cuddle most of the time, do a little exploring of the living room (never went up the stairs or to other parts of the house) and just be awesome. They even knew to go potty in their cage and would go there to do the deed. When they were tired, they'd put themselves to bed in their rat hammock in their cage.
If you get rats, you need to get at least TWO. They are highly social animals that need another rat to snuggle, groom, and tussle with.