This may be old news, but it was news to me. Hope it saves you lots of grief.
I had stripped the Phillips head on a screw in my gear shift mechanism. And yes, it happened because I was impatient and didn't wait until I could find the impact driver. How many times do I have to learn this lesson?
Anyway, there was no way to cut a slot in the screw head, and the existing Phillips slots were too buggered for the now-found impact driver to work. I was about to drill out the screw and then at least have to clean out the threads with a tap. More likely that I would mess up the threads in the block and have to redrill and retap the hole.
In an inspired moment (thanks, Tecate), rather than drill out the entire screw, I drilled a hole about 1/2 the diameter of the screw (the screw, not the head of the screw) and about 1/3 the length of the screw. I hoped that this hole would give the point of a Phillips impact bit somewhere to go rather than just bottoming out before its sharp edges could engage with the screw head.
A couple of whacks on the driver and it came out beautifully, in one piece. In the picture you can see the 4 marks from the impact bit.
Pointers: 1) Don't make the hole any bigger than 1/2 the screw's diameter. This way you leave plenty of metal and you're less likely to just twist the head off. 2) Use a relatively pointy bit, to insure that the point can go down the hole a ways, enough to let the edges of the bit engage.