It use to be possible to find a domain registrar that offers free private whois registration, but this is becoming increasingly more difficult.  Name.com offered this free in the past, but at the time of this writing costs $1.99.

One of the cheaper options available at this point for a private domain registration is Domainsite.com where I currently pay $.50 for a year of private whois, though they don't make this information easy to find on their site.

So, if you're sick of Godaddy's rather expensive fees for private whois, do some hunting and see what you can find.