To demonstrate this, I went through the typical doctor's office paperwork craziness, and conveniently left off my SSN. I finished my information including name, address, consent signature and all that, just left off my social security number.
Guess what... no one noticed! The doctor's office was able to process my medical insurance, bill them, and finish their end of paperwork without actually needing the SSN they required on the forms. I can't say I was shocked... as I often see information being asked for that no one can explain except to say "Well, we need it for our records".
What I recommend to you, my readers, is the next time you're at your doctor's office filling out one of those forms... skip the SSN. See if anyone notices. If they ask you why you skipped it- question them and ask them specifically what the SSN is required for. Don't take lame "we need it for billing" as an answer.