As an American citizen, you are taxed on your worldwide income regardless of residence. So while you can easily set up an LLC or corporation somewhere else, it will not be any different in terms of tax which ultimately affects how much wealth you can store.
Under FATCA, any bank account you or the company holds abroad will be reported back to the US. Can't hide money.
Your best options to lower your tax burdens are to change how your income is received (capital gains rather than salary) or take the big leap of obtaining a different citizenship and renouncing your American citizenship, and leave the US.
The first might be doable if you set up the right arrangements with a lawyer/trustee. Speak with firms like Ogier, Conyers, Appleby, and Walkers. But the fees charged to set something like this up may cost more than your tax savings, so make sure you have a full view of the costs before proceeding.