Taken to assume "100% anonymous" even from governments, those days are gone.
Even if many registered agents in the US don't ask for KYC (at least for forming LLCs), which does mean you can start a company there without providing genuine details, you lose anonymity if you ever want your company to access the financial systems of the world.
Solve your problem in a way that doesn't require anonymity, because that's practically unfeasible nowadays.
I totally agree, one of the best legal methods are to relocate to a country that fits your vision of best life standard and tax planning.
Most people will not be able to relocate, governments know that so actually they have a tight grip on the majority of the worlds populations ball's. There are still options (legal) to reduce you tax wherever you are, most often this is connected with bigger sums to be invested before the setup pays of.
If you want a company which engages in real business like getting paid by and pay 3rd party, then it will be impossible if you don't want to use illegal methods like darks etc. . However there are options with trustees and managed systems to stay anonymous infront of tax. However these setups are more expensive and come with quite some restrictions.
But yes, the best way is to solve it in the 100% legal way in which you relocate.