I'm 34 and from Australia. I don't know if any clapping games are still being played by the young kids, but I've been trying to remember the ones I played when I was in primary school.

Miss Mary Mak:
We had the same version that everyone has mentioned here, though I could have sworn we said that the cat jumped so high, not the elephant. Plus, one of my cousins had the surname of Mak, so I loved playing it whenever I visited her.

The hand clapping went like this:
*With two girls facing each other)*
         Miiiss
(hands crossed in front, touching own shoulders)
         Maaa-
(own hands touching your knees, straight down)
         -ryyyy
(clap your own hands together)
         Mak!
(slap your right hand only with the other girl's right hand, in effect crossing your arm in front of you, and then clap your own hands together)
         Mak!
(slap your left hand only with the other girl's left hand, then clap your own hands)
         Mak!
(slap your right hand only with the other girl's right hand)
         All dressed in black, black, black
(repeat hands movements)

Etc. until the end of the song, going faster and faster, repeating the song until someone fumbles.

Ronald McDonald:
We sang a slightly different version, but I can't, for the life me, remember it.

Under the Apple Tree. This was our version:

Under the apple tree
My boyfriend said to me
Kiss me, hug me,
tell me that you love me

Under the apple tree
My boyfriend said to me
Get lost!

Chinese Checkers:
We sang a different version. I think this is how it went:

I went to a Chinese restaurant
To buy a loaf of bread
I asked him what his name was
And this is what he said
Myyyy naaaaame is
Chinese Checkers
Cheese on toast
Put it in the oven
And poke, poke poke!

I remember a couple of games too. One was Crocodile, crocodile. My school playground had a couple of huge tractor tyres, laying down and embedded in the ground. Us kids would stand on top of the tyre, with one kid standing in the hole in the middle, designated "it". Then we would chant this song:

Crocodile, Crocodile, can I cross the river?
If not, why not, what's your favourite colour?

"It" would shout a colour (i.e. Pink!) and any kid wearing that colour tried to jump to the other side of the tyre without being tagged by "it". If tagged, that kid became "it" and the old "it" joined the other kids on the tyre for the next chant. And so on.

Another game I played with my cousins was "Red Light, Green Light". There were a lot of us in roughly the same age group, so this was a popular game for us at get togethers.

All of us lined up at one end of the back yard, this was the starting line.

One kid, designated "It", went up to the other end of the bark yard (the finish line) and had their back turned to the rest of us.

"It" would start the game by yelling out "Green Light!" which meant that you could start walking up to the finish line. You weren't allowed to run.

It would then yell out either Green Light! or Red Light! at any moment.

If It yelled Green Light!, you were safe to keep creeping to the finish line. If It yelled Red Light!, you had to freeze so that when he/she spun around, you weren't caught moving and had to go back to the starting line and creep up again.

To make it harder knowing which was Red Light or Green Light, It would half turn whenever yelling out "Green Light", trying to trick us into actually freezing. You could freeze, actually, but that meant other peeople were overtaking you by not freezing. You had to pay attention to words, not actions.

The first person to get to the finish and tag It was the winner.