It looks like the front top manifold stud is broke, I can hear the exhaust leak the first 20 seconds of running from a cold engine.

How much damage could I do just driving like this? Any idea how much I am looking at to have the stud replaced figuring the manifold is okay?

'07 explorer 4.6L

 

The longer it leaks, the bigger the chance of it burning a path in the manifold, causing it to need to be replaced too.
I haven't looked at the 4.6L in an Explorer to see ho difficult that it would be to remove the manifold to replace the stud, so I would be afraid to guess the labor time involved.
The dealer should have upgraded stainless steel studs available.