There are a couple of others I like:

See saw, Marjorie Dawe
Johnny shall have a new master
He shall earn just a penny a day
Because he can't work any faster

---

Sing a song of sixpence, a pocket full of rye
Four-and-twenty blackbirds baked in a pie
When the pie was opened the birds began to sing
Wasn't that a dainty dish to set before a king?

The king was in the counting house, counting out his money
The queen was in the parlour, eating bread and honey
The maid was in the garden, hanging out the clothes
When down came a blackbird and pecked off her nose.

---

The Wise Man built his house upon the rock
The Wise Man built his house upon the rock
The Wise Man built his house upon the rock
And the rain came tumbling down.

The rain came down and the floods came up
The rain came down and the floods came up
The rain came down and the floods came up
And the house upon the rock stood firm.

The foolish man built his house upon the sand
The foolish man built his house upon the sand
The foolish man built his house upon the sand
And the rain came tumbling down.

The rain came down and the floods came up
The rain came down and the floods came up
The rain came down and the floods came up
And the house upon the sand fell flat!

---

The Owl and The Pussycat went to sea
On a beautiful pea-green boat
They took some honey and plenty of money
Wrapped up in a five-pound note
The owl looked up to the stars above
And sang to a small guitar
"O lovely pussy, o pussy my love, what a beautiful pussy you are, you are, what a beautiful pussy you are."

Pussy said to the owl: "You elegant fowl
How charmingly sweet you sing.
O let us be married, too long we have tarried
But what shall we do for a ring?"
So they sailed away for a year and a day
To the land where the bong-tree grows
And there in the wood a piggy-wig stood
With a ring at hte end of his nose, his nose, with a ring at the end of his nose.

"Dear Pig, are you willing to sell for one shilling
Your ring?" Said the piggy: "I will."
So they took it away and were married next day
By the turkey that lives on the hill.
They dined on mince and slices of quince
Which they ate with a runcible spoon
And hand in hand, on the edge of the sand
They danced by the light of the moon, the moon, they danced by the light of the moon.


(Sadly you can't sing Col. Fazackerly unless you do it to the tune of The Fresh Prince of Bel Air, and that's just not that appropriate.)