Compass: Yes, GPS can get you to within a yard of where you want to go. And it will do that unfailingly unless its batteries die, or you're under heavy tree cover, or unless the satellites are off shopping somewhere. Compasses work, period.
Parachute Cord: At least 50 feet of parachute, or 550 cord. You can use it for everything from tying a dead deer to a tree so it won't slide down a hill as you gut it, to lowering your rifle from a treestand. I know one hunter who rappelled down a cliff with parachute cord, thereby saving himself from death by exposure.
Takedown Cleaning Rod: A takedown cleaning rod of the proper diameter for whatever rifle you're using. This is to clear stuck cases, or worse, stuck cartridges, from the chamber. Tap gently for best results. It beats the hell out of a stick, assuming you can even find a stick that fits.
Three Means of Starting a Fire: Who knows what's going to happen out there? I carry waterproof matches, a spark-striker, and a butane lighter with a clear reservoir so I can be sure it's full. To have tinder on hand, saturate half a dozen cotton balls with petroleum jelly and carry them in a watertight container.
Three Small Flashlights: At least one should be of the headlamp variety. Why three? Because at least one is almost guaranteed to crap out on you in the course of a hunt, and at the worst possible time.
Swiss Army Knife: People will laugh at all its blades…until they ask to borrow it. There is hardly any problem this knife will not solve.
First-Aid Kit: A small one will do. You can go for years without using it, and then some day you'll need it very badly. Check it at the start of each season to make sure that nothing needs replacing.
Rubber Gloves: If you like blood all over your hands, you can skip this. However, if you have an open cut and get deer blood in it, you will probably change your mind after said cut gets badly infected.