My exterior door is sagging slightly and won't close as tight as it should. To fix this, I want to tighten the door hinge. Problem is, the screws are as tight as they can get. I can't tighten it any further. Any ideas how to fix?
TL;DR Use Bigger Screws
You can use longer screws, if the frame is deep enough for them, or thicker screws, or screws that are both longer and thicker. Take out one screw and, if you don't have a good collection already, take it to a hardware store to find appropriate, bigger, replacements.
There is also the old toothpick trick - stuff a bunch of toothpicks in the existing holes, with some glue to help them stay put. After the glue dries, chop off the ends so that the surface is level with the frame. Then put the screws back in.
There is also the possibility that the frame is rotting or otherwise simply incapable of holding any screw very well. But unless that's the case, longer or thicker screws should do the job.
This problem is all too common and is usually caused by the top hinge actually bending.
The temporary solution is to remove that top pin and bend the hinge back.
A better solution is to replace the hinge or all hinges with one made of stronger metal.
The best solution is to convert to a 4-hinge door with strong hinges to begin with. It's practically irresponsible that manufacturers don't do this by default, especially for heavy exterior doors!
I recommend watching this video to properly diagnose the problem before taking the wrong action: https://www.youtube.com/watch?v=ZcDj6g2UODw