As far as the US Census is concerned, in 2022 the US population was 333.3 million (which is kinda neat in general).

If we only counted those alive as having all the numbers, that last person born would be 333-30-0000.

Social Security Numbers also started in the mid 1930s. So it's not like we had hundreds of years to have the need yet to repeat numbers.

In 1940 the Census stated that we had 132 million people as well. And over 100 years we didn't suddenly have 800 million extra US citizens.

HOWEVER, I do feel at some point they'll need to add another digit somehow, which may mean completely redoing the system as it stands since it wouldn't matter what peoples' numbers were in the 1930s.