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How can something so small have more surface area than a basketball court?
there's no magic here. For example, activated carbon [1] has a surface of 500 m^2 per gram. It's just a very porous structure.

[1] http://en.wikipedia.org/wiki/Activated_carbon

an easier analogy: 100-stories building that stays on 1 cm^2 area has a total surface of 100 cm^2. Now, make the height of this building 1 cm, and you have 100 cm^2 surface in 1 cm^3 volume.

Make it 1M floors, and you are talking about 1M cm^2 = 100 m^2

Imagine a sheet of paper the size of a football field but infinitesimally thin. Now, fold it in half. Fold it in half again. Continue folding it in half until it's the size of a postage stamp. You now have something with the surface area of a football field that you can easily fit in your wallet.
fun fact: there are a few miles of metal interconnect (wires) inside the microprocessor chip you are using right now [1].

[1] www.itrs.net/Links/2007ITRS/2007_Chapters/2007_Interconnect.pdf

I've been hoping for a supercapacitor laptop battery for some time. I'll gladly take a 3 hour battery life if I can get a 30 second recharge.
Where would you get the high-amp power supply though?
It doesn't have to be a high-amp power supply- you could use a low-amp supply to charge up another supercapacitor, then use that capacitor to charge your laptop's supercapacitor.
the best thing about supercapacitors is that they don't have a liquid inside. It means that (at least theoretically) they can be of any form and thickness. So, (again, speculatively speaking) they can be a part of laptop frame.
> they don't have a liquid inside

They do:

"...electrolyte (an ionic liquid called EBIMF 1 M)..."

My fault. I thought they use the approach described in [1], where the prototype used LightScribe DVD optical drive to make a pattern on a disk with graphite oxide film. That prototype didn't require any liquid.

[1] http://www.sciencemag.org/content/335/6074/1326

Not sure if I want to carry that kind of high power energy source in my bag, or on my lap... http://www.youtube.com/watch?v=EoWMF3VkI6U
I don't see how this video shows anything which supports your .. fears? Obviously, you shouldn't short-circuit capacitors. You shouldn't short-circuit any power source.
The big question is how long they can retain the charge. Supercapacitors generally have a much higher self-discharge rate than batteries.
I think the first usage would be to recover energy from the brakes, energy that you can use almost immediately afterwords. You would still use another battery as the main storage.
Note to reader: the supercaps in the article aren't being used for long-term energy storage.

When a car brakes using magnets, a huge quantity of power is generated by the magnets. In a typical setup, this power value is much higher than the maximum rate at which the lithium batteries can absorb charge.

From what I gather, what this article actually means is that these supercaps will be used as energy couplers between the regenerative brakes and the lithium batteries. They aren't actually replacing the lithium battery.

Yeah, like a 1st level cache for energy.
So is the idea here that you would use these supercapacitors as the primary energy store? Or is the case that you would use them to quickly capture the energy from regenerative breaking and then use it to slowly charge traditional batteries over time?
I think the idea is that you capture the energy from regenerative braking in the super capacitors and then drain them almost immediately when acceleration is called for (which means the batteries need to be used a lot less and thus last a lot longer).
Put it in my phone, tablet and laptop ... if I have something that I could charge for 16 seconds I don't care how much it power source will last.

Also super-capacitors aren't that bad for medium term storage - they lose 10% of the charge per day which is not much.

I don't know much of the science, but I think if it can charge that quickly, it can probably discharge pretty quickly as well. I imagine I wouldn't want to be on the receiving end of that, so maybe there are drawbacks to putting it into a device you hold in your hands?
The lenovo x1 carbon (and probably others) charge from 0% to 80% of battery in about 30 minutes -- which is really helpful and does a lot to compensate an otherwise lackluster battery life.
Sounds interesting but how exactly is 64Wh/kg "almost comparable" with 100-200Wh/kg?

Personally, I'm all for supercaps for regenerative braking right now and will love to see a day when we can fully charge an EV in 30 seconds.