Sorry about my last post, it didn't go through right. I'm 18, so these are fairly good recollections, although I was never in the inner circle so some of these are a bit secondhand. I'll put the chant/rhyme in quotes, and then the performance instructions and such below that...

"Little Sally Walker,
Walkin' down the street
she didn't know what to do so
she stopped in front of me she said
Hey girl, do your thing, do your thing and switch
Hey girl, do your thing, do your thing and switch."

A group stands in a circle and Sally skips/walks/runs around the circle. at "She stopped in front of me" Sally picks a girl, stops in front of her, and during the first "Hey girl" bit does some kind of action. The ones I've seen ranged from cheer leading herkies, to dance moves, to jumping jacks. Then the girl who was stopped in front of switches places with Sally and repeats the action, and becomes the new Sally as the chant starts over again. We played this on our last day of 8th grade in Kansas, the year was 2006.
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A similar version, but more appropriate for both genders, which is why we switched to this one on that same day, once the boys wanted to join in and preferred not to be referred to as Sally.

"Ridin' ridin' ridin' my pony
ridin' ridin' ridin' my pony
ridin' ridin' ridin' my pony
and this is how it goes.
Front to front to front with my pony
back to back to back with my pony
side to side to side with my pony
and this is how it goes."

So during the "ridin" bits, the person gallops around the circle, stops in front of someone at the first "And this is how it goes" line. The rest is fairly explanatory...rub fronts, rub backs, rub sides, and switch places.
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I always loved this one!! Girl Scouts all over have used this one for ages, as far as I can tell. some girls moved from Pennsylvania and when I tried showing them this one, they knew a variation on it.

"Down by the banks of the hanky-panky
Where the bullfrogs jump from bank to banky
Singin' ee uh pop puh eye uh pop puh oh uh pop puh pow!"

So the players sit in a circle, criss-cross-applesauce (is that term still in use??) with their knees touching. Everyone placed their hands palm-up on their knees...where two hands overlap, the right hand always goes on top and the left hand beneath it, both palms up. At the beginning of the chant, the first person smacks the palm to her left. That person slaps the next palm, and it continues around. Who ever is slapped on "Pow!" loses. There are variants on what to do with the loser...if the circle is large, they go in the center, but if it's smaller they can just sit outside. This continues until two people are left. These people hold each other's right hand, like shaking hands. They chant and move their hands back and forth (this is difficult to explain. The best way to illustrate it is to clench your right hand into a fist. Bend it horizontal at the elbow. Then punch straight out and back a few times.) If your hand is closer to you at the end, you lose. If it's closer to the other person, you win.
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This was a fun one, guaranteed to drive any bus driver or parent insane. For that reason, you could hear it for 50 percent of any long bus ride, especially the ones with only girls, like volleyball trips.

"There was a hole
in the ground,
the prettiest hole,
that you ever did see.
(quickly)the hole in the ground and the green grass grew all around all around, the green grass grew all around.
There was a tree,
in the hole,
the prettiest tree,
that you ever did see.
(quickly)the tree in the hole and the hole in the ground and the green grass grew all around all around, the green grass grew all around.
There was a limb,
on the tree,
the prettiest limb,
that you ever did see.
(continue with the same theme...now there's a limb on a tree in a hole and green grass)

Continue this theme for as long as humanly possible. Sometimes you inserted bark on the tree before the limb. For the fullest list I've ever come up with (this is in order of their verses)
**hole, tree, bark, limb, branch, twig, nest, egg, bird, feather**
and by the end you are going insane trying to remember everything.
**the feather on the bird and the bird in the egg and the egg in the nest and the nest on the twig and the twig on the branch and the branch on the limb and the limb on the bark and the bark on the tree and the tree in the hole and the hole in the ground**
During the main part of the verse, just sort of chant/sing, the tune is dum dum dum dee, dum dum dum doo, dum dum dum dee, dum dum dum doo. Then quit trying to sing and just say it as fast as you can. It's really funny.
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Another song to annoy...this is one we didn't dare sing on bus trips, because at least the tree song has an end. Don't let small children see this, especially before a long trip.

"I know a song that gets on everybody's nerves, everybody's nerves, everybody's nerves
I know a song that gets on everybody's nerves, and this is how it goes." (repeat to infinity, or until the driver pulls over)

No idea where it originated. But so easy to annoy with!! I'm just at the border of understanding the annoyed parent and the giggling child, so I find it annoying but hilarious.
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Simplest jumping rhyme ever...

"Cinderella
Dressed in yella
went upstairs to kiss her fella
made a mistake
kissed a snake
how many doctors did it take?
1,2,3,4, etc"

As far as I can remember, the jumper's feet hit on the first syllable, and hit on every other syllable (CIN-der-EL-la DRESSED in YEL-la WENT up-STAIRS to KISS her FEL-la) and once you got to numbers, you counted the times the jumper's feet hit. So the jumper hit pavement sixteen times before the numbers.