Generally you incorporate off-shore for one to many of these three reasons: tax minimisation, privacy and corporation structuring.
For most people, an LLC in New Mexico, US (provided you are able to enter the US) with a parent company in the BVI is the easiest option.
Overall this will cost you less than $1,000/yr to operate as an LLC with none of your identification or documentation on it, bank accounts and services available in a first-world country, however operating as a proxy corporation for a BVI-based company therefore not subject to corporation tax.
However there are risks associated with this setup, plenty of people have had bank accounts frozen or assets seized temporarily if their accounts are included in a "probably tax evader or drug cartel" report. So it's suggested to never keep funds in your US-based corporation. Use a reputable, US-based accountant for your annual filing to minimise this risk. If you're not a US citizen or resident, the only risk is the possibility you won't have access to your funds for a long period of time (and you'll likely have to fly back to the US to sort things out).
Now, what I've written above is technically legal provided you're not a US citizen, or if you aren't obligated by law to pay federal or state taxes. However, it - and many, many similar structures are used every day for corrupt, illicit and morally bankrupt reasons.
Nobody is interested in fixing this. We've put in Anti-Money Laundering and Know Your Customer regulations whilst simultaneously PROVIDING ways to make sure people can still get away with tax evasion and hiding dirty money (US: insane protection for corporate entities, UK: owning and operating some of the largest tax havens in the world). As the world begins to wise up to these methods, we're already seeing other super-powers begin to offer the same and in some cases much better services. Taiwan and China, I'm looking at you.