Just to be silly, here's one to the tune of Hava Nagila:

Happy, Happy-a-Birthday, Happy-a-Birthday, Happy-a-Birthday to you!
Happy, Happy-a-Birthday, Happy-a-Birthday, Happy-a-Birthday to you!

Now tell how old you are, now tell how old you are,
Wishing you many more birthdays for you!

(I just made that up. Warning, it could get repetitive too.)

There's also a round sung a lot in the Sacred Circle Dance community, I think it originated in Findhorn:

To you we sing
And happiness we bring
To celebrate your birth
An angel here on earth