How can I remove a Philips screw with a stripped head?
NO. Don't hammer a flat head screwdriver into the thing, hoping to damage the head more, but in a way that will allow you to remove the screw. It won't improve the situation, and it will just damage the screwdriver! Using a hammer like this will only (temporarily) relieve your frustration.

There are many varieties of "easy-out" or "grabit" screw extractor bits sold. Usually they will fit into a drill, allowing you to back out a damaged screw.

If I don't care if I damage the wood around it, I'll just break/pry out some of the wood around the head, enough to be able to grip the head with a pair of vice grips. For example, I have an old beater chisel, used expressly for this purpose. I don't care if I damage the edge, although I probably would not do so as long as I was careful. Or use a Dremel moto-tool to remove the wood around the screw head.

Of course, if you have a moto-tool, then another option is open to you. Use a slotting bit to grind a new flat slot into the head of the screw. The slot will now accept a flat head screwdriver.