I took my dog hiking many times throughout his life. The dog had his own pack which carried most of his food and other items listed below.

Extra water - A dog, depending on it's energy level and fur thickness can go through a lot more water then a human.

Collapsible water bowl - In many areas, there are not many natural water sources and dogs don't like the idea of you sharing the Nalgene bottle with them.

Food and snacks - A medium to large dog in top physical condition can carry around 25 % of their weight. For a 50 lb dog that can be around 10-12 lbs. My dog carried his own food, however I also carried a couple meals worth as you never know when the dog could lose it's pack (yes it happens).

Leash - Invaluable in bear country. Many times we were in very close proximity to bears and having the ability to control your dog so he does not have a confrontation with the bear can be a lifesaver for the dog and you...

Dog boots / paw protectors - These boots are a must have in rocky and hot country. As tough as dog feet may be, they are not immune to paw injury and dog boots are a must. Don't cheap out and buy the fashionable, cheap cotton type that your neighbor forces her precious lap dog wear. There are many protectors designed for working dogs are worth the investment.

Sleeping pad - Yes, a sleeping pad. Depending on the time of year and the breed of the dog, they are also susceptible to cold transfer through the ground. A small and very lightweight, folding pad such as this fits in a backpack and not only provides some comfort for the dog when needed, they are also extremely useful for humans when you need to sit for awhile. Even though we no longer have a dog, these pads are still carried in the backpacks.

Dog sweater - Again some dogs can benefit from a sweater in colder climates.

Dog waste bags - Pickup after your dog. Nothing worse to be on a hiking trail and finding dog mines every 50 feet.

Small towel - Many uses for dogs. Cleaning mud/crude from fur, wiping feet before allowing the dog in the tent etc.

First aid kit - Did not carry anything extra nor dog specific in the first aid kit, however extra rolls of gauze are useful for wrapping small cuts on the dog's legs etc.

Dog toy - Like humans, dogs like to have leisure time. I often brought along a tennis ball or the dog's favorite kong.