OK.
This is the formula for calculating time dilation
where:
t = time observed in the other reference frame (the astronaut in the spaceship)
t0 = time in observers own frame of reference (Earth)
v = the speed of the moving object
c = the speed of light in a vacuum
I put this in a spreadsheet and played around with various values for the speed of the moving object.
If I have it right, where the speed v is 0.99999999999999 c, the spaceship takes t0=10 minutes according to an observer on Earth, the astronaut would experience a time passing t=0.0000849 seconds, well within the capabilities of a 1960s era atomic clock.
In an imaginary thought-experiment type of situation, of course you can push the speed up to any arbitrarily big fraction of the speed of light and of course the time experienced by the astronaut will shrink to an equally arbitrarily small duration. It may exceed the ability of any imaginable clock to resolve, but it won't actually go to zero. It can't.