There are a few things you should read and understand before you read my answer to this question at the bottom of this response. Today Banks and Credit Card companies will try to bully you into giving a Social Security Number. Some have even begun to state that the Patriot Act requires them to obtain a SSN. This however is NOT TRUE.
Below is the Only Passage within the entire Patriot Act that mentions the phrase "Social Security".
[ Patriot Act ]
SEC. 326. VERIFICATION OF IDENTIFICATION. (b) STUDY AND REPORT REQUIRED- Within 6 months after the date of enactment of this Act, the Secretary, in consultation with the Federal functional regulators (as defined in section 509 of the Gramm-Leach-Bliley Act) and other appropriate Government agencies, shall submit a report to the Congress containing recommendations for-- (1) determining the most timely and effective way to require foreign nationals to provide domestic financial institutions and agencies with appropriate and accurate information, comparable to that which is required of United States nationals, concerning the identity, address, and other related information about such foreign nationals necessary to enable such institutions and agencies to comply with the requirements of this section; (2) requiring foreign nationals to apply for and obtain, before opening an account with a domestic financial institution, an identification number which would function similarly to a Social Security number or tax identification number;
If you go to the Social Security Administration government website you'll find that Social Security is a Voluntary program, and only the SSA can require you to give a Social Security Number for the purposes of identification. No American is required to pay into this federally setup Social Security Insurance program, Nor is any American required to apply for or obtain a SSN from the SSA.
That Does Not mean US citizens do not have to pay income taxes.
While it is true that article 16 of the United States Constitution was never properly ratified by Congress, into law, making taxation legal, the federal government has accepted and now enforces income taxation as being true. Therefore all tax codes supported under article 16 come into play.
However nothing within the tax code requires a United States Citizen to have a SSN. The tax code does require US citizens to obtain an Individual Taxation Identification Number in order to pay income taxes, if they do not have a SSN.
A US citizen can obtain an ITIN by contacting the federal Internal Revenue Service. If the IRS asks you for a SSN, tell them you do not have
one. Be prepared to provide your US Passport Number.
You can obtain a US passport without a SSN.
To obtain a US Passport without a SSN you will need to mail in a completed DS-11 US Passport Application form with fees, and you will need to include a reissued "Long Form Original Certified Birth Certificate With A Raised Seal". If any part of the DS-11 form asks you for a SSN it is legal under federal guidelines established for the purposes of obtaining a passport to put "000-00-0000." Feel free to call the US federal government's passport division to double check.
If you have a SSN, that your parent's got for you, and now you don't want one use the link below to find out where to mail a letter of SSN revocation, to terminate your SSN.
Most states will allow you to obtain a Drivers License without a SSN if you bring a letter sent to you by the SSA stating you are not eligible for a SSN. Which I can assure you, you will get if you terminate your SSN number.
As for the Credit Card Companies and Banks.
It will be an up hill battle. You'll more than likely have to call or go in person. If they attempt to deny you:
 Inform them you have terminated your SSN legally in accordance with 20 CFR 3 A7 404.1905, or you do not have one period.
 Inform them that the bank can not be held legally responsible by anyone for failing to obtain a SSN from you pursuant to 31 CFR 103.34(a)(1).
 Inform them that under the Internal Revenue Code Section 6041, they are not required to provide any taxpayer identification numbers on the Form 1099 that they file with the IRS at the end of the year.
 Inform them that pursuant to 26 CFR 301.6109-1(c) they are under no legal federal or state obligation to obtain a SSN from you, and doing so is actually against federal SSA policy.
 Inform them that 42 USC 408 makes it a Felony to use threat, duress, or coercion to try and force a person by fear or deceit to provide a SSN.
They will probably tell you they have to consult their legal team.
With the exception of being told I don't have enough gather-able credit history to offer a line of credit. Every time I have used the aforementioned method with a Bank or Creditor it I've been issued a back account or credit card within 72 hours.