Note, I am not an accountant, but these are my experiences running a business in the U.S.

If you are a passthrough entity like a sole-propietorship, partnership, or S-corp, probably not. Note, this doesn't matter if you're an LLC or not because the IRS doesn't care. Basically, passthrough entities calculate all business income minus deductible expenses as personal income. Therefore, the amount is subject to federal and state income tax as well as payroll tax (FICA). Certainly, income tax is tiered and FICA caps out at $118,500, but let's assume that you're pulling in $100,000. This mostly puts you in the 25% tax bracket. We'll assume that there's also 5% of state income tax. For FICA, since you're both the employee and the employer, you pay 15.3%. All together, that means you're losing about 45% to tax off the bat. Note, this taxes apply regardless of where you register your company unless you live out of the country for 330 days a year or can claim foreign residency. Then, it gets complicated because a certain amount of income is tax free, but not over a certain amount. Anyway, if you decide to live in the U.S., that 45% or so of tax from above applies regardless of where the company is registered. The reason people register their company outside the state where they do their business, other than liability and tort concerns, is because they're trying to avoid sales or gross receipts tax. However, everyone just bills that amount to their customers anyway, so it's really a wash.

If you really want to know the implications where you live, just hire an honest accountant for an hour and ask. That's what I did. It's worth the money.

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Edit 1
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Look, if you really want to save money on taxes, just make more money. That sounds silly, but it's true. Register as an S-corp. Your first $118,500 are brutal because you're paying an extra 15.3%, but after that FICA goes to 0%. Now, the top tax rate in the U.S. is 39.6%, but dividends are how people cheat that rate. Basically, an S-corp is required to pay the people who work in the company the prevailing wage for that position. However, money above that amount can be distributed to the shareholders as dividends. These are taxed at 15%. Note, you can't just underpay yourself and then claim everything as a dividend. That's illegal and you will get caught eventually. That said, the overall tax rate for someone who's running an S-corp and making $300k is almost certainly lower than someone making $100k. For example, if you can convince the IRS that the prevailing wage for your position is $120k, and you made $300k, then the first $120k gets taxed at 45% or so (see above) and the last $180k gets paid out in dividends and taxed at 15%. That's how you get your tax rate down. Note, at this point, really, just hire an honest accountant who can do the correct calculations and paperwork.

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Edit 2
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Note, I talked about being an S-corp above. However, even if you register your company out of the country, it doesn't matter. Americans must file their taxes every year regardless of where they live. If you are a bonified foreign resident, then the first $100,800 is tax exempt, but we still have to file this number. We also have to file what bank accounts have foreign assets over $10k at any time during the year. So, basically, when you file your taxes, the IRS doesn't care where your company is registered at all. You're going to pay your income tax and FICA, which is going to be in the 40% range if not more. Quite simply, if you're self employed, you're paying 15.3% in FICA. If you try to cheat this and your employer didn't pay the other half, the IRS will still want its money and you'll get caught. Even if you setup a foreign bank account with a foreign business, if a company pays you more that $600, they're probably going to file a 1099 in order to deduct that expense from their taxes, which means that the IRS knows about your income. Further, if you try to wire the money into the U.S. from your foreign bank, that goes through a check as well. Really, the IRS wants to get paid and they will one way or another.

It's not that illegal action can't hide money. Certainly, it can. However, it's difficult to hide money or reduce tax liability unless you have a lot of money to pay someone with a lot of know how to do it for you. At that point, just be happy that you're rich.