Want an animal with more fur than face? Consider an English Angora Rabbit like this one pictured. They are nice, friendly creatures with only one main concern; grooming.
These cute not-so-little tribbles need to be shorn every ninety days lest they succumb to a condition called "wool block" in which they injest more wool than they can digest. Shear them regularly and feed them lots of quality hay so that PETA doesn't show up on your doorstep launching vials of fake blood at you and your children.
Oh, and, oddly enough, Angora rabbits are good for people with allergies as they do not carry the same allergens as other rabbits and pets. Take your pick between the English, the French, the German, the Satin, and the Giant. Who wouldn't want a giant rabbit? The bigger, the better eh?