Whose time? As I approach the speed of light (relative to you), even though what you see tells you that my clock is running slow, it’s actually *your* time that appears (to me) to be slowed down (by a factor of [math](1-\frac{v^2}{c^2})^{-1/2}\approx 1+\frac{1}{2}(\frac{v}{c})^2 )[/math].

The speed needed for either of us to notice the effect depends on how accurately we can measure time intervals. But if we can do it to one part in a million (ie to 6dp accuracy) then for us to notice the difference would require [math]\frac{v}{c}\approx 10^{-3}[/math], so roughly a few hundred km/sec. But if you wanted to notice it on anything not moving fast enough to escape the solar system then you’d need a couple more sig figs of accuracy (ie about 0.1 sec in a year). And if you want to see it on a low orbiting satellite you’d need another couple of places (ie to notice about one millisecond delay over a year).