I don't believe that a person can be "too young" to have a Social Security number. Rowan, I bet you have a SSN but possibly you don't know it (assuming you're a US citizen.) You may have slipped through the system if you were born at home with no involvement of a hospital or an MD; otherwise you'd have been registered.

Those of us who are relatively old didn't get numbers assigned to us until we went to work at our first jobs, but for the last several decades, kids born in the US get SSNs assigned to them at birth. My kids, now ages 29, 27, and 24, all have had SSNs since they were infants. To list them as dependant children on my tax return, I always had to fill in a blank for each kid's SSN.

I think the assignment of numbers was restricted to welfare recipients at first, but by 1977 they were "numbering" all births, or at least all births in hospitals. We were not receiving any kind of public assistance when any of our kids were born. (Never did, until briefly in the immediate aftermath of Hurricane Katrina.)

Maybe a person could exist in the US without a SSN if:
1) over 30 (or more) ~ that is, born before they system began registering infants;
2) they've never had a "straight" job with payroll deductions;
3) never left the country (requiring a passport), or come here from another country (requiring a "green card").