My grandmother taught us:

"1, 2, 3 O'Lairy
My first name is Mary
My second name is Anna and
That's how you spell Mary Anna

1, 2, 3, O'Lairy
I spy mistress fairy
Sitting on a huckleberry
Reading the dictionary"

I don't actually remember if it was her who taught us the second verse, but that just came back to me as I was typing.

I always heard it as "alairy", but that's how she did it, always while bouncing her leg over the ball. We were actually just talking about it the other day.