We used to say one that went:

Say Say my Playmate, (clap, clap)
Come out and play with me (clap, clap)
And bring your dollies 3 (clap, clap)
Climb up my apple tree (clap, clap)
Slide down my rainbow (clap, clap)
Into my cellar door (clap, clap)
and we'll be jolly friends (clap, clap)
Forever more, more (clap, clap)
10-4 (clap)

Then there was:
Apples, Apples on a stick,
make me sick,
make my tummy go 2-4-6
not because it's dirty,
not because it's clean,
just because I kissed a boy
behind a magazine,
boys, boys have some fun
here comes the teacher with her mini skirt on
she can wibble, she can wobble
she can do the splits,
but i betcha 5 dollars she can't do this,
close your eyes and count to 10
and if you mess up start over again,
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
the end!

Then there was one about strawberry shortcake... can't remember that one at all for some reason.

then there was about billy boy
cross down when billy boy was 1
he learned to suck his thumb
thumb after thumb after half past one


2 was tie his shoe
3 was climb a tree
4 was shut the door
5 was swim and dive
6 was pick up sticks
7 was pray to Heaven
8 was shut the gate
9 was pay the fine
10 was say the end

and at the end it was
... end after end after half past ten
cross down the end.

then there was one about
cinderella dressed in yella
went upstairs to kiss her fella
made a mistake kissed a snake
how many doctors did it take?
1, 2 and so on until you missed at jumping rope