I'm looking for the self-proclaimed "last private web hosting" site, that requires no name. You could send them $1 USD in the mail with a desired account & password & they would set it up for you.
You'll need a 24/7 box somewhere, but it could even be behind a residential NAT with only outbound connections (at reduced performance than having a public port). You can easily & transparently move it from one physical place to another; your hash identifies it, not the physical routing location.
I2P includes a webserver, to host the local configuration & status page. You can also host your own public site right from there, by dropping files in the right place. Click "Website" from your console page and it'll tell you what you need to do. The URL will be <some hash>.i2p, but you can register with one of the name servers to give it a more human-friendly <name>.i2p redirect, similar to DNS->IP address.
"If you live outside the United States and can demonstrate that the site you wish to host would put you at significant, legitimate risk of retaliation from a government with a documented track record of reprisal against people who speak out against it, we may be able to help. Anonymous hosting is serious business; it can be one component of a coordinated plan to protect you and your family from torture and murder. It's absolutely not an option you can use to dodge lawsuits or unpopularity arising from hosted material.
If you feel you need this level of protection, please contact us, taking appropriate privacy precautions with respect to your correspondence. Be sure to explain where you live, what you want to host, and why you feel hosting the material anonymously is the only way to guarantee your safety. Be very specific; you will need to explain your situation in enough detail so we can make an informed decision. We may, in our sole discretion, decide to waive the contact information requirement in exchange for periodic reviews of your site content by NearlyFreeSpeech.NET personnel to verify that your usage of the service is consistent with your claims. Please be aware that even if we approve your request, paying anonymously is extremely difficult.
We regret we are not able to provide anonymous hosting to residents of the United States under any circumstances."
I created my account and log in through Tor and don't experience any problems. I don't think I ever logged in outside of Tor, but I don't consider it to be a security requirement and am not sensitive about it, so it's possible there are exception (in particular it's that I misremember creating the account through Tor).
I built something that starts instances with Bitcoin, if that helps any: https://www.stackmonkey.com/. You add ssh keys with a callback, so the site never sees who you are. The virtual appliances run on my HP Cloud account, so for now your instance would be there.
Will be adding container tech to it at some point.
That's not anonymous with the amount of google cruft on there. At least google will know that you instantiated something there and google is within the US so can be subpoenaed to reveal the details of whoever visited your site.
Note how the sdf page linked elsewhere in this thread is completely clean in that respect. They do have some other issues but at least they try to keep third parties out of the loop.
This is not what the user requested but it's definitively worth for the attention they put on privacy and activism. The tech collettive is totally autonomous. Here is the Manifesto http://www.autistici.org/en/who/manifesto.html