A GPS unit isn't mandatory, but it's extremely useful. That said, it can be replaced by an app on a smartphone. There are several that can store maps and tracks offline. I use MyTrails (about $3), but there are others just as good.
In general, a smartphone is one of the most useful things you can bring. Phone, internet, camera, music, GPS, all in one small device. If you keep a blog, the PackJournal app allows you to type your journals, save them offline, and upload them later, and the website shows you where you wrote them and took your photos, even if you were offline at the time. Only needs one charger, and you can probably find a waterproof case, which isn't true for most other electronics.

I second Dr. Bronner's soap. Liquids are heavy. This gets you down to one, and it's concentrated, so you don't need to bring much. You can even use it as toothpaste (not pleasant, but it works). In bear country, using the unscented one as toothpaste is a great idea.

The importance of a waterproof/breathable jacket and waterproof panniers/bags can hardly be overstated. Even if you don't expect rain, waterproof/breathable jackets are among the best things for cold weather.

Other than that:
Tent
Sleeping bag
Sleeping pad
Spare tube, cables, patch kit, tire boot
Good multi-tool
Sunscreen
Toothbrush