An article in The Economist (see original article) describes how an Australian researcher named Jason Sharman tried to register anonymous companies and open bank accounts for them in 45 places around the world. He was successful in 17 cases, 13 of which were OECD countries, including the USA and the UK.
In these cases, the bank can't reveal the owner of an account to any inquisitive government, because the bank doesn't know. So these accounts are safer from investigation than those in a traditional tax haven, where personal identification is normally required.
The USA is particularly attractive in this respect because there you can apparently get tax-free interest on your untraceable riches.
I pass on this useful tip to any of you who may be wealthy enough to consider taking advantage of it. However, bear in mind that you shouldn't believe everything you read, even in The Economist. There may be hidden snags.