Which bank lets you open a checking account without a social security number?

Actually None. A Social Security Number is a mandatory requirement if you want to open a checking account in the United States. However, if you are a new resident of the country (say a foreign worker) the bank will allow you to open the account with the SSN but you have to apply for your SSN and update your bank records within the first month of opening of the account. If you fail to do so, the bank can freeze or even close your account.