Neat idea. It has crossed my mind too.
IIRC the code is already counting the number of notifications you have not read yet,
Yes, with the current implementation, it would be no more expensive to display the total count rather than a boolean yes/no value. That is because there the smallest API gets all entries in a bucket; there's nothing more lightweight.
As far as the design/usability/UX change, I'll think about it. We can use this thread to track that.
It also reminds me of the original iOS badges, hehe.

Here's how Twitter the iOS app does it:

While it's certainly possible (and inexpensive) to do this, it doesn't seem like something people really want or would benefit from.
Another factor is that the circle is quite small:

So it'd be really hard to fit a legible number in there.
I think a solution I'll go with is to show the number of unread notifications in a tooltip, when user hovers (indicating they might care about additional details):

That way the presentation is still legible and not distracting when not looking at the notifications icon.
A red circle appears when you have notifications. IIRC the code is already counting the number of notifications you have not read yet, so it'd be cool if inside the red circle was the number of notifications that you have (I think the Gmail iPhone app does/used to do this on the home screen, for example)