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In Star Wars (again), one of the most recurring characters is R2-D2. Even though he's in most of the movies, I (personally) have never seen him charging. I've seen him do a lot of power consuming activities, like shocking baddies, but I've never seen him charge. What kind of battery (existing or non-existing right now) is needed for this feat?

As with all things where you need a lot of power stored compactly, I'd argue that R2D2 is nuclear. More specifically, he probably comes equipped with an RTG - Radioisotope Thermoelectric Generator.

An RTG is basically a box of radioactive material that gets hot, wrapped in some form of thermo-voltaic material that converts the heat to electricity. These are not even new technology; some of the first RTGs were put into the Voyager probes and used to power them consistently as they left the light and heat of the sun. The real issue with these devices in current technology is efficiency; They actually quite warm and only generate a small amount of electricity. In space, this is actually a good thing because electronics don't work once you get them below a certain threshold temperature so the warmth does as much to keep everything running as the power does.

But for Star Wars, let's assume that R2D2 has a more efficient thermoelectric generator; some form of thermo-voltaic material that is much better at converting the heat generated by the radioactive material into power. That would give him the power he needs to do most of what you see him doing, but given that the output of an RTG is constant, he probably has a conventional battery as well; this would allow him to store the electrical output from the quieter times to supplement the constant output of the RTG for specific high load tasks. As such, you'd only need a small RTG, that perhaps generates 60% of the power needed during active periods, and the battery stores the surplus during the times when he's not moving or shocking people to supplement his power output for when he is.

I would also point out that battery storage capability is currently going through a massive surge in research and development right now as the world prepares for a future of electric cars and sustainable generation so while it's impossible to know what that supplementary battery is made of just yet, odds are good there's some special rare earth metals involved.

Good thing there are planets like Tattooine out there to exploit for them I guess.

If we assume that the future has much more effective batteries and energy gathering than now (which is probably a good assumption considering the Death Star), then this could be wide open.

He could actually charge himself by movement, absorbing kinetic energy to power his batteries. He could have very tiny solar panels, or a way to absorb other energy straight from the air. He may even be able to take the heat that the constant use of his electronics would generate and use it to power himself.

Heck, considering they have laser technology--solidified light that expires after a certain distance--and light sabers, they almost certainly have more knowledge of science than we do, allowing them to use processes to create and power their droids we can't even imagine. Honestly, we may not know how these droids are powered, as they may be using something we haven't even discovered yet.

But what I think is probably most likely is that we constantly saw him charging himself.

We often saw him plugged in to computers, ships, and other places--it may be that every time he plugged in to do his various tasks, he not only downloaded data but also recharged himself. This may well be something he was able to do in mere seconds.

If his battery is good enough, once charged, there's an excellent chance that he could run for years or even decades.

In the newest movies he comes alive after spending far to long as a rather large doorstop, and Word of God (the creators) was that it took most of the movie for him to reboot himself. That does seem to imply a low battery since the rest of him was working just fine, but even after a decade or three just powered down in the corner, he still managed the juice to do it. That does seem to imply a rather large capacity, especially since he activated himself, which means the entire time some rudimentary process would have to have been running.