@ThePiachu - I agree, the Dot-BIT project is very anonymous...if you want to register a .bit domain that is (only .bit is available from the project so far). The problem with .bit domains is that the system is still very much in it's infancy and so hasn't developed to a point where it is as easily accessed as other "ICANN" domains.
In regards to the question, most registrars will require a certain level of identification when registering a domain (ICANN rules and regs). I know of a lot of registrars which only require a name, email address and physical address. I know, thats a lot of info if you are trying to remain anonymous, but paying by Bitcoin removes the vital registration check that matches the bank details with the other reg. info.
I'm NOT saying that you should lie, because there is a level of trust between registrars and their customers, which would be a shame to see disappear. Also fake info can lead to you losing the domain anyway. But the use of Bitcoin in itself will provide a certain level of anonymity.
If you request whois privacy, the regristrar will ensure that your details aren't available to the public, but they (and ICANN) will still have your details in their files.
As for complete anonymity, I'm unaware of any registrar offering that legally. It would have to be a decentralised domain system, one that wasn't regulated by ICANN, like Dot-BIT...
A really good list of Bitcoin-using registars can be found on this site at Where can I register a domain and pay with bitcoins?
This question appears to be off-topic. – Stéphane Gimenez – 2014-01-06T11:11:34.523