The short answer is “no, not really.”
The longer answer is that you should consult with your CPA, to understand how certain types of income may need to be reported (i.e. do you file state income taxes while abroad or not, and if not, could pass-through income with a company trigger the requirement?).
If you care about anonymity, then you should form a LLC in either Delaware, New Mexico or Wyoming, as I would advise anyone – expat or otherwise. You just need to be careful about the physical mailing address you use. Read more on whether you need to use your foreign address or not when forming a company in the US while living abroad.
If you don’t care about anonymity, then I would recommend you first consider forming first in your state of residence. If that’s not of interest to you, then the next priority would be where you think you would be “transacting business” such that you would trigger that state’s requirements to register the company.