Best state for anonymity
As far as anonymity goes different states have different solutions. Some states are very invasive and make it hard to remain invisible to the general public. Other states have less requirements as far as what information is collected and then published for everyone to see.
I am going to mention a few states that are more discreet than others. Also I will mention nominee service, using third party incorporator/organizer, and how to remain anonymous using a commercial registered agent service (I would limit the scope of this answer to regular for-profit corporations and LLCs, but many of those rules would apply to the other entity types as well):
1. Delaware: In general Delaware does not require listing the names of officers, directors, or shareholders of Corporations, and members/managers of LLCs. The Articles still require a registered agent to be mentioned, as well as incorporator/organizer, so to maintain complete privacy it is necessary to hire third party providers for those services (see 4 and 5).
2. New Mexico LLCs: New Mexico LLC is a popular asset protection tool. Like Delaware, New Mexico does not collect any information about members/managers of LLC (keep in this rule only works for LLCs, not corporations). The relative advantage of NM LLC is lack of annual reports, fees or taxes, and cheap filing fee. This way once formed the NM LLC cannot be administratively dissolved because of maintenance formalities (you are still required to maintain a registered agent though). Similar to Delaware, to maintain complete privacy you need third party registered agent and organizer for the LLC.
3. Nevada, Wyoming, and nominee service: Nevada and Wyoming are two states that alongside Delaware are considered "incorporation heavens" due to combination of favorable laws and other factors. Among other things both states permit what is called "nominee service". The idea is to replace real owners by "nominees" who would appear in public records. One big disadvantage of nominee service is its price - typically it costs much more than an immediately available alternative.
The immediately available alternative is forming a NM LLC to be the holding company of your other companies in states that require listing owners/officers. This way using a NM LLC you can register a company anonymously in practically any state (that strategy works primarily for LLCs).
4. Using third party incorporator/organizer: When you form a corporation an incorporator needs to sign the Articles of Incorporation (similarly, organizer signs Articles of Organization of an LLC). To make sure your name does not appear on the Articles it is common to use third party incorporator/organizer for your company. Once company is formed, an incorporator/organizer would resign and issue a letter of resignation.
Keep in mind that a few states require the owners/officers of the company to act as incorporators/organizers and do not permit third party to act in that capacity.
5. Using commercial registered agent to protect your anonymity: Almost every state require listing a registered agent name and address on the Articles of Incorporation/Organization (the address must be a physical address in the state of registration). Many company owners choose to serve as registered agents for their companies, however among many disadvantages of that choice one is that of loss of anonymity - their name and address become publicly available. The best solution is to hire a commercial registered agent and list their name and address on the Articles.
FYI: The IRS just a side note - no matter how carefully you conceal your true identity from the state this is one agency which requires you as the company representative to list your name (and tax ID - SSN, EIN, ITIN - provided you have one), in order to obtain the EIN (tax ID) for your company. The good news is this info is not made public.
I hope this answer gives you some idea how to protect your identity better using various techniques and choosing the right state.