• Passport - everyone needs this so this is a no-brainer.
• Comfortable Shoes - much of the country's activities revolve around participating in the great outdoors. This means you'll be doing lots of walking and hiking in some rugged mountainous terrain ranging from muddy moors to cliff-hugging granite trails to alpine forests to grasslands. At the very minimum, comfortable walking shoes are a must, but hiking boots are probably better (for ankle support among other things).
• Hat - don't take for granted the dangers of UV radiation. As you're closer to the north pole, where charged particles are funneled by the earth's magnetic field, contracting solar-induced cancer is easier than you think. A hat will at least keep your scalp from getting severely burned. If you're wearing a broad-rimmed hiking hat, it could also help protect your neck, ears, and face.
• Sunscreen - again, given the sun's harmful UV rays, it's a good idea to protect other exposed parts of your skin from sunburn.
• Sunglasses - prevents cataracts or other harmful effects of prolonged exposure to the sun's ultraviolet rays on your eyes
• Bug Repellant - this can be controversial because the chemicals contained in these things can be harmful to the environment. However, mosquitoes can be quite annoying in summer. In any case, the most effective repellants contain DEET, but they're smelly and toxic if used excessively. The better way to go environmentally, though not necessarily a comfortable one, is to cover your skin with long sleeves and pants.
• Layered Clothing - this means wearing multiple layers of jackets, sweats, or shirts and taking them off/on depending on the conditions. This is very important to be able to adapt to the variable weather and climate in the country. It can easily go from freezing cold to sweaty hot and back in a day. This will at least protect you from exposure to hypothermia (even from your own sweat).
• Rain Coat - this will keep you and your equipment dry for the most part (with a rain poncho covering the pack in addition to the body). Given how rainy it can be year round in some parts of the country, this will at least keep you sane under such conditions (not to mention limit your exposure to hypothermia).
• First-Aid Kit - this often-overlooked necessity allows you to disinfect and close wounds or tape up a sprained ankle. You never know when Murphy's Law strikes and having this kit will at least put you in a situation to deal with the consequences rather than letting circumstances degenerate into something more life-threatening than it has to be.

Some other things worth bringing to enhance your experience include...

• Hiking Sticks/Trekking Poles - this is useful for longer and more involved tramping excursions (especially multi-day or very long day hikes). They provide you 1 or 2 extra "legs" to maintain your balance on stream crossings or alleviate shock to your knees if you're carrying a pack
• Lots of Memory or Film or Portable Hard Drive - the first and third items are for digital photographers. In any case, you'll be taking heaps of photos and you'll want to make sure you can bring all your photos home

Finally, nature is inherently dangerous and unforgiving, and since Norway is best enjoyed in the outdoors, you need to be cognizant of these dangers. Below is a list of things you need to be aware of to stay safe in the outdoors...

• Getting lost - always stay on the tracks and heed the signs or trail markers (cairns or spray-painted rocks). Getting off track is a real possibility, especially for longer day hikes where following the trail isn't straight forward. Bring a topo map as well as a compass with you if you're on a more involved hike where it's easier to lose the track.
• Hypothermia - given the amount of high elevation and high lattitude adventures, the danger is very real. Always pay attention to the weather reports before undertaking any tramp that exposes you to the elements. Bring a rain coat to avoid getting wet and losing heat that way.
• Dropoffs - given the rugged mountainous terrain, you're likely to come across dropoff hazards on a trail, road, or even a viewpoint, where a fall can easily be fatal. Many of the granite cliffs are nearly vertical, and you have trails and roads hugging these cliffs. The best advice here is to stay away from the edges to the best of your ability to limit the chances of falling off or having the cliff go unstable on you.
• Road Fatigue - given the long distances and curvy narrow roads, you'll need to be wary of this. To prevent falling asleep behind the wheel or losing your focus, try to switch drivers periodically or break up the drive (rather than tackle it all at once).
• Mountain/Fjord Roads - roads winding their way around fjords and hugging cliffs are common. In fact, you'll be sharing two-way traffic on roads that look like they barely fit two cars side by side. Many locals won't even slow down when they go by you. If all this freaks you out, you can always minimize the chances of actually getting into a wreck by initiating the slow down (or even pulling over) yourself. Moreover, many roads have very steep gradients. This means you'll want to put the car in a low gear when climbing while letting the engine help you while descending. Don't rely solely on your brakes as they will fail!

I'd like to say as long as possible, but this is probably more constrained by how much money you're willing to blow off. We spent about 2.5 weeks on our trip and was rushed, but I did start to tire of spending tremendous sums of money considering how expensive everything was. But even given our limited time and funds, we didn't make it far into Troms county nor Nordkapp county.

If could do it all over again, I'd make better use of the Hurtigruta, which travels the fjords from Bergen all the way to Kirkenes (by the Russian border). Along the way, you could get on and off as often as you want and try to coordinate things so you travel by night and drive off during the day. Of course some serious planning and bookings must be done way ahead of time to pull this off, but this would be a tremendous time saver.

Another thing I'd do over would be to give the trip 3 weeks at least. That way, the whole trip wouldn't feel nearly as rushed as our previous trip.