Nothing's too silly to waste time on! The internet is proof of that! (Just check out the "odd links" thread.)

How about

Little one, lean one, long one, lickpot, jumbo
Count your little fingers as I say
Little one, lean one, long one, lickpot, jumbo
You'll be mama's pride and joy someday.

Might be more words, but I don't know them. It's a finger counting song and we had it one of those amazing bright red records that were made for kids in the fifties...

Also:
Put another nickel in
in the nickelodeon
Come on, baby, loud as sin,
It's music music music

... (oh bother, I've forgotten this verse!)...

Louder, now play it louder
Come on we'll clap our hands and stamp our feet
And when we've got a steady beat
Put another nickel in, etc.