Wim Wenders shot Wings of Desire here. It's a quiet place to work and definitely a gorgeous place to visit. Combine it with a visit to the Philharmonic, just across the street. Affiliation is 30 euros per year and you have lots of desks all over the library to choose from.
The best cheeseburger in Berlin costs less than 5. Burgermeister is mandatory in town when you're craving for a basic and delicious to the bits chesseburger.
There's nothing more revigorating during the dark winter days than spending a couple of hours in a sauna. Germans have mastered the art of sauna and you can find amazing pieces of architecture in the public swimming pools (Stadtbad). I find Stadtbad Neukölln very peculiar in its neoclassic architecture and it's definitely a wallet-friendly option to enjoy some sauna time there. Be prepared to be naked in front of man and woman!