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The third law of thermodynamics states that nothing can reach to absolute zero temperature. What is the lowest possible temperature that can be in the universe? Has any experiment reached to a billionth of a Kelvin? Is there any restriction on how low it can be? Is $10^{-1000}\;\rm K$ possible? Or is there a lowest quantum of temperature?

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An epsilon above absolute zero is entirely fine, re Bose-Einstein condensates. When you state temperature you should also be concerned with entropy. Silicon dioxide as fused silica is not the same as alpha-quartz at "0 K." The first BE condensate ran at 170 nanokelvin. Cooling of Bose-Einstein condensates below 500 picokelvin,

http://cua.mit.edu/ketterle_group/Projects_2003/Summaries_03/Picokelvin.pdf

One can also cool a macroscopic object to its quantum ground state. Last, negative temps kelvin are easily achieved - imaged bodies in an MRI tunnel, running lasers, population inversions overall. One can thus arbitrarily closely approach absolute zero from either side. However - that shalt not cross over! You must go the long way around.

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The temperature of the blackbody radiation predicted from the event horizon of a black hole with mass $M$ is $$ T = \frac{\hbar c^3}{8\pi k G M} = \frac{62\,\mathrm{nK}}{M/M_\text{sun}}. $$ So a supermassive black hole with $M=10^9M_\text{sun}$ would have a horizon temperature in the neighborhood of $10^{-16}\,\mathrm K$.

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