Here's a couple from Oz if they haven't been posted yet. I remeber most of these (actually - ALL of them - - - - sad. I must be getting old(er).

Eeni Meenie macka racka
rare rye dominacka
Chickapop alollipop
Rang pang puss

Usually followed by:

Penny on the water, Tuppence on the sea
Threepence on the whirly-whirly
You're not 'he'

(UK or US kids might say "You're not 'it'")

Another goes

Eenie meenie makka rakka
rear rie damma nakker
rang pang push

Here're a few others:

Eerie oarie ickory am
Queerbie quorby raspberry jam
Filsy folsie Irishman
Tickle em, tackle em; bosh

AND

Intery mintery cuttery corn
Brambly briar and brambly thorn
Wire briar barrel lock
Three geese in a flock
One flew east and one flew west
And one flew over the cuckoo's nest

(which is probably how the bbok and film of that title came to be - a nonsense rhyme and the fact that cuckoos don't build nests)

Ickey ackey horses cackey (poo)
what colour will it be?

(child landed on would nominate a colour - preferably a long one to ensure they didn't get landed on for "it / he" . . . .

Yellow - Y..E..L..L..O..W

Whoever copped the "W" was "it" or "he" for the chasey game.

I went to a Chinese restaurant to buy a loaf of bread
He wrapped it up in a ten-bob note and this is what he said:
"Eli eli chickabye chickabye, saw a sausage - in bed!
Went to the other one, got me another one: Drop dead!"

AND FINALLY

I saw Esau sitting on a see-saw; I saw Esau sitting on a gate
I saw Esau sitting on a see-saw; I saw Esau kissing our Kate

Thes, of course were all 'counting rhymes to dtermine who was going to be doing the chasing in the game about to commence and were generally made as complicated as possible in order to prevent rigging the outcome.

I also recall one that seemed to end with "Bumbarumba gosh - Nineteen Hundred and One!" (1901 being the year Australia achieved Federation status and the right to govern itself independent of English veto oe approval).

Muttley

My wasn't that fun!