As I remember it, the rhyme goes like this:

One, Two, Buckle my shoe (so far, so good)
Three, Four, Shut the door (yup, it rhymes)
Five, Six, Pick up sticks (ok with that one)
Seven, Eight, Lay them straight (got the verb right, too)
Nine, Ten, Big fat hen (alrighty then)
Eleven, Twelve, Dig and delve (is there such a word?)
Thirteen, Fourteen, Maids a-courting (weak)
Fifteen, Sixteen, Maids in the kitchen (not even close)
Seventeen, Eighteen, Maids a-waiting (totally lost it)
Nineteen, Twenty, My plate's empty (the end, thanks a lot)

Do I have the words right? As a child, I always cringed at the almost-rhymes of the teens, and I wondered at the lack of imagination or "ear" or something.

Then there's the alphabet song. I never learned it, nor the alphabet, until I had to teach it to younger children. The teacher was aghast and had the class sing it for me. Instant learning, just add music.

That early exposure to rhymes that don't rhyme and don't scan still didn't prepare me for Donne's poem:

Go and catch a falling star,
Get with child a mandrake root,
Tell me where all past years are
Or who cleft the Devil's foot.

If you're going to write rhyming verse, then make it rhyme already. If you're going to write free verse, then rhyme doesn't matter, but the rhythm still underlies everything:

No man is an island, entire of itself; every man is a piece of the continent, a part of the main; if a clod be washed away by the sea, Europe is the less, as well as if a promontory were, as well as if a manor of thy friend's or of thine own were; any man's death diminishes me, because I am involved in mankind, and therefore never send to know for whom the bell tolls; it tolls for thee.

Now *that* is verse. Yes, I know it is an extract from one of his meditations, but his prose was really often beautiful free verse, and his verse was terrible and should have been washed away by the sea.