Does anyone know if it is possible to open a bank account in the USA
completely online without having a social security number or setting
foot in the US? I am a South African resident and would like to open a
bank account in the US from here. I have tried with some of the main
banks e.g. PNC Bank, Wells Fargo, Citi Bank etc but the social
security number is a required field on all their forms and when I
called the banks they say that it is not possible to do without a
cocial security number.

Dear Andy Higgins, 

"De Jure", there is no legal obligation to have a Social Security
Number: see for example - http://www.no-ssn.com/ . It is also possible
to open a checking account without SSN.

According to this government site, if you're a freigner asked for SSN,
all you have to do is write "foreigner" in the box.
http://www4.od.nih.gov/ofm/faq/questions.stm

I don't know if it will actually work, since, as you discovered, "de
facto" it is a common practice and banks would rarely open an account
for you without an SNN. While searching for a legal answer to your
question, I ran into a considerable amount of libertarian and
anarcho-liberal sites and other people who promote avoiding paying
taxes claiming that legally, banks have no right to ask you for your
SS number (which in your case you don't have). See, for example:
http://www.theedgeoftime.com/bank.html ;
http://www.takelifeback.com/oto/otoh191.htm
http://bradbva.home.mindspring.com/freedom/bankacct.htm 
http://www.greaterthings.com/News/Victories/bank_ssn.htm 

Finally, what's you've been looking for, after all the legal stuff: 
These firms offer to open US bank accounts for foreigners without SSN
- I can't vouch for them:
http://www.valisinternational.com/?go.php=usaccount 
http://www.non-us.com/index1.htm 


Further Reading:
---------------
Hiding Your Money : Everything You Need to Know About Keeping Your
Money and Valuables Safe from Predators and Greedy Creditors
by Jerome Schneider, Allison Hope Weine
http://www.amazon.com/exec/obidos/ASIN/0761523405/ltc-political is a
book that advices how to open a bank account without SS number.
However, I did not read it and some amazon.com reviews in this page
are pretty critical.

http://www.livetaxfree.com/products2.asp - This site also sells a book
with similar advices. Again, I haven't read the book so I can't
recommend it.

Here's a "success story" - a foreigner without SS number who managed
to open a bank account.
http://www.moonrise.ws/Captain/LFA-02.html

Search Terms Used
-----------------
"bank account" without "SS number" 
"bank account" without "Social Security number" 
"bank account" without SSN 
"bank account" foreigners us

I hope that helped a little. Please contact me if you have any further
questions.

Thank you for the answer - I am most satified with it.

You want an International Taxpayer Identification Number ("ITIN"),
which you can obtain without fear of exposure to US tax:

"The IRS has permanent staff which are able to help process Forms W-7
at the following U.S. embassies overseas: Berlin, London, Mexico City,
Paris, Rome, Singapore, and Tokyo. In addition, there are public
accounting firms overseas in certain countries which are acceptance
agents for ITIN numbers. You will find their names and addresses
Acceptance Agent Program page.
If none of the above persons can help, then the IRS will accept a Form
W-7 accompanied by ORIGINAL documents which establish to the IRS
     the identity of the foreign person, and 
     the foreign status of the ITIN applicant. 
The best documents to provide for this purpose would be a foreign
passport and/or a foreign national identity card. The IRS will accept
copies of documents establishing foreign identity and foreign status
if the copies have been certified by the government agency (foreign or
domestic) which issued the documents, employees of the U.S. State
Department located in U.S. embassies and consulates abroad, or a U.S.
notary public who is allowed under state law to notarize foreign
documents.
In addition, there is an alternative procedure which can be used by an
alien who resides outside of the United States to certify copies of
documents attached to his Form W-7 so that he does not have to attach
original documents to such Form W-7 when he mails it in to the
Philadelphia Service Center to apply for an ITIN from abroad."

International Taxpayer - Obtaining an ITIN from Abroad 
http://www.irs.gov/businesses/small/intltaxpayer/display/0,,i1%3D2%26i2%3D23%26i3%3D37%26genericId%3D79883,00.html