Purchase third party whois privacy. Whois privacy costs nothing to run, it's just a company saying they own the domain on your behalf, this opens you up to domain theft though, I don't understand why people use it. Set up a PO Box somewhere and use that.
3.7.7.1 The Registered Name Holder shall provide to Registrar accurate and reliable contact details and promptly correct and update them during the term of the Registered Name registration, including: the full name, postal address, e-mail address, voice telephone number, and fax number if available of the Registered Name Holder; name of authorized person for contact purposes in the case of an Registered Name Holder that is an organization, association, or corporation; and the data elements listed in Subsections 3.3.1.2, 3.3.1.7 and 3.3.1.8.
3.7.7.2 A Registered Name Holder's willful provision of inaccurate or unreliable information, its willful failure promptly to update information provided to Registrar, or its failure to respond for over fifteen (15) calendar days to inquiries by Registrar concerning the accuracy of contact details associated with the Registered Name Holder's registration shall constitute a material breach of the Registered Name Holder-registrar contract and be a basis for cancellation of the Registered Name registration.
From what I've seen (I have personally never done it, but know people that have), this is not typically enforced in any way, and you're probably fine entering completely fictional information for all of this stuff.
But...
I knew a guy in college (super disreputable, FWIW, so I don't necessarily believe everything he said) who claimed that one of the first tricks up his sleeve when he was launching a new site was to file complaints about incorrect WHOIS data against all his competitors (and sometimes against random sites with useful keyword content and a bit of pagerank) and then pounce on the domains if they opened up so he could point them to his new site. He told me that often enough to be useful to him, people either failed to respond to the verification requests in time, or had actually provided false data, and he was able to either snatch their domains or at least cause some trouble.
So I probably wouldn't recommend messing around with phony data, better just to set up some contact info that you don't mind having out in the open.
Which essentially means that you cannot transfer any domains out of GoDaddy without losing your anonymity.