I don't know all the requirements from all the banks in the U.S. but the ones that I know will ask you to provide a U.S. address.
However, there are banks, mainly in Florida as someone has mentioned, that will open an account for you without a U.S. address.
I have an account with Chase and I don't live in the U.S. but Chase will now ask you for a U.S. address.
But you have two solutions that I can think of...
1) You open an account with a U.S. Broker that besides giving you a brokerage account will also give you a checking account, and a debit card. I recommend Charles Schwab, but you'll need to provide an address somewhere in the world.
2) The second option is that I put in contact with someone that will open an account for you in either Chase or BofA without having to provide a U.S. address, but you will have to provide an address somewhere in the World (the guy that I know is legit, I know him, but he will charge you a few thousand Euros, he offers a lot of services related to Banking, Company formation, etc.)
I also know a company that does provides the same services as the guy that I've just mentioned, the company is in Singapore and they are also legit. They will charge you of course.
But regardless of the path you choose to open an account in the U.S. you will have to provide an address somewhere, maybe your parents home if you don't want to give your address.