The first SSNs were issued in 1936 https://en.wikipedia.org/wiki/Social_Security_number
According to the death master file entry in wiki 111x10^6 SSNs died between 1962 and 2018. https://en.wikipedia.org/wiki/Death_Master_File
That's 1.982 x 106 x deaths x year-1. Assume that number to be a constant during the period 1936-2024
1.982 x 106 x deaths x year-1 x (2024-1936) x year = 174.4 x 10^6 deaths
According to https://en.wikipedia.org/wiki/Demographics_of_the_United_States there's 335.9 x 10^6 residents, but I can't tell if they are citizens with SSNs, but I'm going to assume that for now.
So (335.9 + 174.4) x 106 is 510.3 x 106 spent SSNs.
According to the same demographics wiki article the birth rate is 11 births per 1000 population. Death rate is 10.4 deaths per 1000 population. Because I'm just doing back of the envelope estimation for fun, while trying to manage my hangover in the early afternoon, I'm not going to create an exponential function to describe population growth. Instead I'm going to only consider future the US population a constant and not consider the 200 x 10^3 annual net growth (it only affects the next year's growth by 120 anyway)
With all of that BS out of the way, at the present birthrate the US requires 3.695 x 106 new SSNs annually. The total amount SSNs in the current scheme is (109) - 1. I'm going to be leaving out the -1. 109 total SSNs - 510.3x6 spent SSNs leaves 489.7 x 10^6 SSNs available. 489.7/3.695 is 132.5.
So in conclusion, assuming a constant population, the US can go for another 132.5 years with the present scheme without having to reuse any SSN.