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It all depends on the amount of 9s.
At 99% speed of light you age 8.5 seconds per every minute. At 99.9% it's 2.7 At 99.99% it's 0.85 At 99.999% it's 0.27
At 99.99944444% it approximates what you described, with 1 second becoming 5 minutes. But you can keep adding 9s, at 99.99999% one second becomes over half an hour, and at 99.99999999999999% it becomes over 2 years.
So on and so forth, so at 99.999...% like you said it's essentially 0 seconds vs infinite time.
You can play around with this calculator to get the numbers https://www.omnicalculator.com/physics/time-dilation