I am in a position where I am able to work from any location of the world. The dozens of clients I work with (currently as a simple self-employed) are scattered throughout the globe, and I work only via Internet.
I am interested in moving to a LLC form, for peace of mind.
After a life spent in a sort of prison - long story - I feel the need to move from country to country every year or so. Yet, I would like to have an "official" base for my business somewhere in the world, and a place where to pay taxes/pension fund and so on.
Are there some countries that you would pick as a base for yourself?
You're going to be in a tax nightmare. Many countries will require you to report income and pay taxes just by mere residing there for long periods (=months). Your company may be required to qualify as a foreign company while you're there. You might be liable for salary withholding, you might be liable for corporate taxes.
Different countries have different laws, but the one thing they have in common is that they do like to tax as much as they can. Foreign persons should also consider tax treaties and the laws of their country, for example if you're a US citizen - you will be paying taxes to the US government regardless of where you are.
This is something you need to consider with proper legal advice. Those stories that you keep hearing about off-shore personal pocketbook corporations - people who do that can certainly afford a lawyer or two on a retainer.