I originally took the test and obtained my license in 2008 with just my resident permit. Don't have a SSN and not going to get one because I'm a conscientous objector to a lot of federal fascism and want no part of it.
Then became a citizen, changed my name, updated license in person, SSN question did not even come up.
When I later renewed my license I had to sign another line affiriming that I don't have a SSN. Peachy.

This seems to be the prevailing sentiment about SSNs:
http://apps.leg.wa.gov/RCW/default.aspx?cite=26.23.150
Of course, a lot depends on the office that you visit. None of that extra paperwork was needed in my case.

To whomever is reading this, please only move here if you are a libertarian and have trade skills to support yourself.
So tired of the californian demagoguery and populist BS about "jobs". The only "job" you can have is the one you create.