For example, Goldilocks and the three bears, the three blind mice (not sure if it's a fairytale though!), the three pigs, the three fairies in Sleeping Beauty, etc. etc. Also, from other more obscure fairytales (welsh, celtic, etc), most families have three children. I was rewatching Shrek, and was just curious haha.
Most Helpful Guy
I'm no psychologist so I don't have a well informed background, but I would say it's because three is a safe number. It's not too many to keep track of, but not as intimate as just two. Two individuals often times indicate a special bond; two lovers, Hansle and Grettle, mother and father, the hero and their rival. Three is less intimate and more stable.
In terms of building and construction (my line of work), triangles or threes are much stronger snd more stable than things with 2 or 4. A three legged stool with never wobble the way 4 legged chairs in elementary school always seem to. You and a cross member to turn a flimsy square frame into two triangles that won't colapse. Maybe there is some subconscious realization that three is stable and balanced and that translates into the three bears, the three faries in sleeping beauty, Ron Harry and Hermione in Harry Potter, the three witches in MacBeth, the three dates of Greek myth, etc1
Most Helpful Girl
Those fairytales were mainly made by Christians and the number 3 is a holy number in Christianity. Like the legend of saint Nicholas who put 3 fingers in the air and a miracle happend.1