I was taking a wild guess. 

Waiting sunnyfrutti to answer this question.

It's confusing now.  Some sites say you cannot open most checking account or get a credit card without a SSN; others say SSN or ITIN is required to get a credit card.

Her is another one that I found is rather interesting:

You are not REQUIRED to disclose your SS# to get a credit card, open a bank account, etc.

under the Social Security Privacy law, a bank can not deny you privileges and rights because you refuse to disclose a number, which only identifies a government trust fund established for an individual. Social security numbers are only for that purpose, for the businesses of the the Social Security Administration.

In small claims court, you can, under this Privacy Act, sue and rcv a guaranteed $1000 for each offense. The majority of low level bank employees and managers do not know this, but most executive level employees do.

What you do is send a professionally written letter to the bank vice president, including the statement that this letter is being sent under the Privacy Act laws regarding Social Security numbers, and that you are seeking the right to open up a bank account at his/her institution, and would like him/her to set up an appointment with one of the bankers to open up a bank account at his bank. The executive will naturally brief the banker about alternative methods like using dummy numbers (which many banks have at their disposal), call you to set up a time, and you get your bank account. If they refuse, going to small claims court is cheap and requires no lawyer and you are guaranteed by this Privacy Act of monetary damages. In nearly all cases, the bank will back off.

With credit card applications, in the entry for your SS#, write SEE ENCLOSED LEGAL NOTICE, and submit the application along with a copy of the Privacy Act. Send by Certified mail along with Return Receipt. Some credit card companies will call you back and tell you what documentation and proof of income you will need to send to them, including references, employer statements, copies of W-2 (with SS# scratched off), etc. You might get lower than average credit limit but it works. The rest of the companies will deny your application, at which you either can send a leagl warning letter (nicely yet tactfully) explaining that you will sue for the Privacy Acts guaranteed award amounts in small claims court if they do not allow alternative means of proving ability to pay. Some of the card companies will back off. The rest of them you will make $1000 each offense.

Also, if any institution backs down after you file suit, follow through with the suit to get their penalties and then get your credit card.

A better way to avoid these procedures is to file for an ITIN from the IRS. You will have to build some credit history on the new number before getting a credit card, but you can get a bank account right away.