Wow that's great, there are loads on that link!
I will add some more
1. Twinkle, twinkle, little star,
How I wonder what you are. [sometimes it's where you are]
Up above the world so high
Like a diamond in the sky
Twinkle, twinkle, little star
How I wonder what you are.
2. Row, row, row your boat, gently down the stream
Merrily, merrily, merrily, merrily, Life is but a dream.
3. I went to the animal fair
The birds and the beasts were there
The big baboon by the light of the moon
was combing his auburn hair
The monkey fell out of his bunk
And slid down the elephant's trunk (wheeeee!)
The elephant sneezed and fell on his knees
But what became of the monkey, monkey, monkey, monkey, monkey, monkey monk?
Twinkle, Twinkle little bat
How I wonder what you're at
Up above the world you fly
Like a tea-tray in the sky.
The original is a poem by Jane Taylor,
The Star (1806) and it has four more verses. The other is, of course, the Mad Hatter's, which he started to sing at "the great concert given by the Queen of Hearts" but she screamed he was "murdering the time" (killing it not being sufficient

) and she declared (as usual) "Off with his head."
The usual tune is known in French as: "Ah, vous dirai-je, Maman?"
Mozart wrote a quite delightful set of variations on it.
Here is one set of words:
Ah ! Vous dirai-je, Maman,
Ce qui cause mon tourment ?
Papa veut que je raisonne,
Comme une grande personne ;
Moi, je dis que les bonbons
Valent mieux que la raison.
But here is another (and probably earlier) set, just to show that Carroll did not get his ideas about "fair use" from thin air:
Ah ! Vous dirai-je, Maman,
Ce qui cause mon tourment ?
Depuis que j'ai vu Silvandre,
Me regarder d'un air tendre ;
Mon cœur dit à chaque instant :
« Peut-on vivre sans amant ? »
L'autre jour, dans un bosquet,
De fleurs il fit un bouquet ;
Il en para ma houlette
Me disant : « Belle brunette,
Flore est moins belle que toi ;
L'amour moins tendre que moi. »
Je rougis et par malheur
Un soupir trahit mon cœur.
Le cruel avec adresse,
Profita de ma faiblesse :
Hélas, Maman ! Un faux pas
Me fit tomber dans ses bras.
Je n'avais pour tout soutien
Que ma houlette et mon chien.
L'amour, voulant ma défaite,
Ecarta chien et houlette ;
Ah ! Qu'on goûte de douceur,
Quand l'amour prend soin d'un cœur!
And the version of "The Animal Fair" I know has:
The monkey! He got drunk
(Alternative: The funniest was the monk, Who sat)
And SAT on the Elephant's trunk.
The Elephant sneezed etc.
Which makes more sense, since the monkey was sitting, probably, on the end of his trunk and the sneeze blew him to Kingdom Come! The author of this one is unknown. It has a charming tune as well.
And "Row, row, row" is a song meant to be sung as a round (a four part canon).
Adding some others which also have tunes:
1. Round and round the mulberry bush (the cobbler's bench)
The monkey chased the weasel.
The monkey thought 'twas all in fun.
Pop! goes the weasel.
A penny for a spool of thread,
A penny for a needle,
That's the way to money goes.
Pop! goes the weasel.
2. Jack and Jill went up the hill
To fetch a pail of water.
Jack fell down
And broke his crown,
And Jill came
Tumbling after.
3. Three blind mice!
Three blind mice!
See how they run!
She how they run!
They all ran up to the farmer's wife.
She cut off their tails with a carving knife.
Have you ever seen such a sight in your life,
As three blind mice!
(this one can also be sung as a round...)