Chosen Solution

I have a PS3 Slim in which I was going to replace the hard drive on, however the blue screw to get to the hard drive enclosure is so incredibly tight that even with the most amount of force I can muster, I cannot, for the life of me, turn this screw even a bit. Any suggestions?

Try putting a towel with a lot of grip in between your hand and the screwdriver, hopefully the towel will increase traction between you and your hand. If your driver keeps slipping on the screw, stop trying to loosen it because you will strip the screw. Find a place that has the proper screwdriver for the job and take it there.

I would take it into a hardware store and make sure you have the correct bit. You might need an impact driver or just more brute force.