Updated this post with some new info:
I didn't know this one, but it seems to have been fairly popular in the US, the UK and Australia over the last thirty or forty years, at least:
My boyfriend gave me an apple
my boyfriend gave me a pear
my boyfriend gave me a kiss on the lips
and threw me down the stairs
I gave him back his apple
I gave him back his pear
I gave him back his kiss on the lips
and threw HIM down the stairs
There are several third verses, mostly involving going to the movies (since all this stair-throwing didn't end the relationship):
He took me to the movies
to buy some bubble gum
and when he wasn't looking
I stuck it up his bum
Then there are (almost inevitably) underwear variations:
I threw him over London
I threw him over France
I threw him over Harbour bride
he lost his underpants
(This is often followed by a verse where he flies all over London, France, and Harbour Bridge (or the USA or any number of other "third" places) to find his underpants.
Folks, this is one messed-up relationship. Some folklorists say this is about girl empowerment, but I'm not really buying it. This is a couple that throws each other down the stairs then goes to the movies to engage in all kinds of deviant acts - it's either a real BDSM power couple or a seriously dysfunctional couple with no one I can really sympathize with (something tells me that that initial stair-throwing wasn't the first act of violence in the relationship).
But all analysis aside, how old is this? It's been appearing in print since the 1980s, but almost certainly goes back further than that - a variation was published in Captain Billy's Whiz Bang in 1921:
First I gave her peaches
then I gave her pears
then I gave her fifty cents
and kissed her on the stairs
This sort of suggests that it may have come from ribald rhymes about prostitutes - by the 1970s or 80s, though, violent rhymes were probably less likely to get you in trouble than sex rhymes (at least in the states, where violent media tends to be much more likely to be considered "family friendly" than sexual media).
By way of trying to put in a centuries-old tradition, maybe we can also connect it to a couple of old, old folk songs:
The first of these is "I gave my love a cherry, it had no stone." "My boyfriend gave me an apple" sort of follows the pattern of that one (and is considerably less boring).
But a more obvious, and far more likely, thing would be to connect it to the old Irish standard "Do You Love An Apple"
Do you love an apple?
Do you love a pear?
Do you love a laddie
with curly brown hair?
Yes, I love him
and can't deny him
I will follow
wherever he goes
This is essentially the same song without the violence - the playground version sort of follows the song to a natural conclusion of what might happen if you promise to follow a laddie wherever he goes (ie, he could turn out to be a real jerk).