I've made a number of piñatas. For ages 3 to 12. And believe me, making a piñata a group of 12 year olds don't destroy in a single swing can be a challenge.
For your 3 year olds, you're only going to want a few layers of mache. Anything more than about 5 and they'll get frustrated.
THE HARDEST THING IS MAKING THE STRING STRONG ENOUGH/NOT RIP OUT. Especially with a full piñata. My solution has been to wrap the strings around the entire piñata.
The tombstonata:
7-8 year olds. Base was a cereal box (left inside). about 10 coats of mache. string runs around the edges.
The unicorn.
Very dead (didn't get a 'living' pic) Shoebox base, neck/head and legs just for looks. 5 coats mache Ages 4-5. String around the shoebox.
Traditional star work in progress (no completed pic):
Ages 5-12, about 30 kids were going to be swinging at this. Basketball base, made 2 half shells, then mache them together. The points are party hats. From here, I wrapped dental floss around the ball, added a layer, wrapped in another direction, added a layer, etc. Overall 15 layers or mache. The rope holding the piñata broke, not the string or piñata. Every kid got to swing, without so much as a blindfold before it finally gave way. I used to have a video, but it's lost.