Peter, Peter, Pumkin eater,
Had a wife and couldn't keep her,
Had other, and couldn't lover her
Up the chimney he did shove her

Or
I'm a little tea pot short and stout-
Here is my Handle, here is my spout
(look down at arms, and find them both akimbo)
Oh, shit, I am a sugar bowl!

I learned both of those before the age of 10-
and later this one...

Old Mother Hubard, went to the cupboard,
To get her poor doggie a bone
When she bent over, the dog got a boner,
and gave her a bone of his own...

and there is the "a weela weela walla" the Dublins childrens song, that i never really learned, and only remember a a word or two...

about the mother who stuck a knife in her baby's head (a weela, weela walla, down by the river sallya)
(and the results--in all the gory legal details)
and Moral of the story is, don't go sticking knifes in baby's heads (a weela, weela walla, down by the river sallya!)
(the song is always sung in a cheerful way-- even though it is somewhat morbid!)