It's amazing to me that there's like ten responses here and not one of them is even remotely encouraging. Here's a dose of reality tinged with optimism: you will not get a new country unless you start a civil war somewhere, win, and make it one of the conditions of peace that you get a new country. But what you can get, which is even better (because it saves you from having to care about issuing currency/license plates/passports/etc) is a special economic zone. The idea is that this an area of the country where there are reduced taxes, simplified business registration, and less opportunities for bureaucratic shakedowns. In the Dubai International Financial Centre they even use English common law instead of UAE law. In order to encourage investment the countries will say something like "the rules in this SEZ will not change for at least 50 years."

Closer to what you are looking for is a subset of SEZs called charter cities which usually have the goal of changing much more than just the rules on commerce. Próspera in Honduras is an example of this but unfortunately some legislative changes have sort of left it hanging.