There are some states that have laws that will provide you with a certain amount of privacy, but you may face difficulty running a business entirely anonymously if it is based in the US. The reason is that shareholders / members of a business typically have to be disclosed for tax and other purposes and an owner can not hide from those responsibilities by remaining anonymous.
But, there are some steps you can take to maintain a reasonable amount of privacy. Jurisdictions like Delaware or Wyoming, for example, have laws that will allow you to run the business without your name directly in the public view. There are ways to continue the anonymity but you have to be strategic in going about it.
I would be very careful trying to set up a business anonymously because there are a lot of laws that need to be complied with. For that reason, I suggest that you speak with an attorney before trying to set up your business. An experienced attorney will be able to walk you through your privacy options and help you develop a strategy. If you still have questions, I encourage you to visitwhere we will match you with a top attorney for a free consultation to answer your questions.