I'm an US citizen who's thinking about going to university in Europe-mostly likely Germany. I would like to eventually live and work somewhere in Europe. Possibly in the UK as I have some family & friends there. I know Germany doesn't allow dual citizenship, but is there a residency permit that would allow me to live/work in the EU or the UK? I know the UK is a bit different on their immigration than the rest of Europe. Sorry I've tried looking everywhere for an answer!
First of all note, that while both Germany and the UK are EU Member states, the former is part of the Schengen area, while the latter is not, which might complicate things if you want to move from one country to the other. (A Schengen visa would allow you to do hassle free short-term travels to other Schengen countries). Note however that for US citizens both the UK and Schengen countries allow you to enter the area visa-free for short term stays of maximum 90 days.
I assume you'll want to learn in Germany then move to the UK to work. In this case this is what you'll need to do:
Note that since Germany is a part of the Schengen area while you study there you are free to do short-term travels (maximum of 3 months in any 6 month period) to other Schengen Area countries, but not to the UK or Ireland. You might still be able to enter those countries, as an US citizen you do not need a visa, but you should check before travelling.
If you want to move to the UK after graduation you have to get a separate work visa (if you want to work there). As a fresh graduate you will probably need a Tier 2 General visa as a skilled worker, which will be valid for 3 yeras and can be extended for another 3. An other option if you really want to move to the UK is to continue your education there (Masters or PhD).
German citizenship is something that you probably won't have to be concerned about for some time (certainly not just to study there). There are a great number of people around the world who live and work in a country other than their country of citizenship.
The German government has a web page for Migration to Germany which provides all the information you would need regarding the visas required for a stay to study in Germany.
It is possible for you to obtain permanent residency (called by different names, such as "residence title" in Germany, and "indefinite leave to remain" in the UK) in your country of choice. There are application fees and such a permit usually requires a steady job, at least at first.
SztupY already provided an excellent answer regarding the procedure in Germany and the UK but to answer your question directly: There are no residence permit allowing you to live and work everywhere in the EU because immigration/long-stays is still by and large a competence of the member states. The only thing close to it is the European Blue Card but it's quite restrictive and has not been very successful. The UK is not part of it in any case.
Once you have a residence card from a Schengen member state (again not the UK), you can however visit all of them for short stays (you might even get away with cheating a little as there are no systematic border checks and no stamps on entry/exit). On the other hand, if you want to move from one country to the other and you don't have a European Blue Card, you have to start from scratch and you will be treated like someone coming to the EU for the first time. Finally, if you are a US citizen, visiting the UK while being based in a Schengen country should not be a problem as you do not need a visa (as US citizen).
Finally, citizenship is a complex topic but if you are able to get one (UK by naturalization for example, perhaps Portugal or Italy based on ancestry?), then none of this matters anymore. EU citizens have a right to work and settle everywhere in the EU. It's perfectly possible to get a relatively “easy” EU citizenship that does not preclude dual citizenship in your situation and then use that to live in a stricter country like Germany.