I wonder in which circumstances one can open a bank account without a Social Security number in the USA.
I'll just add that it is more complex than saying, "there's no SSN requirement". Maybe on paper this is true, but as you will soon find out that is not the case at a lot of banks.
Speaking from experience (wife had no SSN at the time - 2 - 3 months ago) I was unable to open her an account at a lot of banks (Etrade, TD Ameritrade, Ally, Capital One 360, BofA, and Wells Fargo). I was also unable to add her as a joint account holder on any of my accounts (Etrade, Betterment, Capital One 360) so the best I could do was add her as my beneficiary.
PNC Bank was willing to let her open an account without SSN. You just need a passport and another form of ID (drivers license or something). I ended up not doing it because she obtained her SSN card, but if you have a PNC bank near you they will let you open one. I can't speak to all the others obviously. I would call and ask them, or visit the ones close to you.
It's worth "checking" though in case it is different from state-to-state. I was in Virginia.
From my experience, banks require an SSN or an ITIN (Individual Tax-payer Identification Number). If your situation bars you from being assigned an SSN, you might want to visit the local Social Security Administration and ask if you can apply for an ITIN, most people are eligible. The process usually takes about 2-3 weeks. I have not yet come across a bank that will allow you to maintain an account without these. I think banks have some form of reporting that they do to the IRS, because your account might bear income, and that's where the identification number becomes necessary.
The Schwab page to open a brokerage account indicates that the SSN is required to open a brokerage accounts due to IRS regulations and the USA PATRIOT Act:
All brokerage firms require this information for new account applicants to comply with IRS regulations and the USA PATRIOT Act.