There’s a high likelihood that you would find anonymous web hosting if you look in the right places. There are basically two options;
If you require a basic blog which should be free and anonymous at the same time, then you could use Tumblr or WordPress.com. You should ensure that during registration you provide an anonymous email address and also hide your true IP by connecting through a reliable VPN or Tor.
This option is preferable but you would need to avoid EU, U.S, and UK companies because they are constrained by local laws and for a majority of them, domain registrars are required by ICANN to maintain a WHOIS database that is available to the public.
You will discover soon enough that your identity is usually requested by U.S and EU domain registrars before a domain can be registered for you. It is mandatory that you pay by credit card and you also have to bear in mind that your contact information will be public record. You will discover that there’s only a short list when you start searching for reliable registrars and hosts.
Additional Security Considerations
If you run an anonymous website, then you should be avoiding using Google Analytics on the site. You also need to stay away from all third-party services that could be used to uncover your identity. Use your anonymous email account and a VPN or Tor when you send files to your site via FTP or when getting in touch with your anonymous web hosting provider.
Your domain cPanel will give you access to create a webmail account but it is recommended that you use an anonymous email which is different from your hosting and domain. If you plan to trade goods anonymously, you will have to accept bitcoin.