Can one be applied for without getting an SSN?
According to the DS-11 form and instructions a US citizen is required to provide one if they have it. It is possible to fill out the form without one:
Section 6039E of the Internal Revenue Code (26 U.S.C. 6039E) requires you to provide your Social Security number (SSN), if you have one, when you apply for or renew a U.S. passport. If you have not been issued a SSN, enter zeros in box #5 of this form. If you are residing abroad, you must also provide the name of the foreign country in which you are residing. The U.S. Department of State must provide your SSN and foreign residence information to the U.S. Department of Treasury. If you fail to provide the information, you are subject to a $500 penalty enforced by the IRS. All questions on this matter should be directed to the nearest IRS office.
One thing to note is that according to the State Department:
Failure to provide information requested on the form, including your social security number, may result in significant processing delays and/or the denial of your application.
So if you choose to apply for a US passport without one you should have a valid reason why your child doesn't have a Social Security Number and needs a US passport.
So just yesterday I went to request for my newborn's passport. I didn't* have his SSN. I told them it hasn't arrived yet and I don't have it and they were like "no problem, nevermind, it's not needed" as if it didn't matter at all.
I actually went and applied for express in a non-USPS office. They delivered the passport to you in less than 8 days...
*: I actually did go to a Social Security Administration office and got his social security number, but still didn't have the card. I offered the officer to correct his SSN number (I had already printed it with 000-00-0000), he again acted as if it wasn't needed but still said "here, write it down".
Knowing that he knew I was applying for 8 days express and have an international flight, his casualness about the SSN made me think it really isn't important after all.
AFAIK it's not important for newborns/minors. I don't know about adults or older children.