Lowest Michaelis constant Km
Take this generic reaction representing the Michaelis-Menten enzyme kinetic:

You can derive the Michaelis-Manten constant $K_m$:
 (for derivation see https://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics)

The quasi-steady state hypothesis was used, therefore the rate of formation and breakdown of $ES$ are the same; this is a good approximation if the enzyme concentration is much less than the substrate concentration $[S]$ or $K_m$ or both. That means that $K_m$ can be extremely small (almost zero) and the Michaelis-Menten equation still be valid proven that the enzyme concentration is much less than  $[S]$.
$K_m$ is very small if:
$k_r$ and $k_{cat}$ are both very small compared to $k_f$, thus if the complex $ES$ is formed very quickly compared to its breakdown.
In this case the reaction proceeds at its maximum velocity $V_{max}=k_{cat}*E_0$, with $E_0=[ES]+[E]$:

About the theoretical limit on how small $K_m$ can be there are two considerations:
 First, $k_f$ cannot be cannot be faster than the diffusion-controlled encounter of an enzyme and its substrate. This means that $k_f$ cannot be higher than $10^9 s^{-1} M^{-1}$  (ref.https://www.ncbi.nlm.nih.gov/books/NBK22430/#:~:text=This%20rate%20cannot%20be%20faster,s%2D1%20M%2D1.)
Second, $k_{cat}$ cannot be too small or the product $P$ formed is produced at a rate not suitable for the cell survival.
Generally, I haven't encountered $K_m$ smaller than $10^{-6}M$ in biological enzymes (e.g Triosephosphate isomerase, TPI) but possibly there are. In TPI, $ k_f$ ∼ $10^9 s^{-1} M^{-1}$, therefore  $k_r$  ∼ $k_{cat}$  ∼ $10^{3}s^{-1}$ that is still quite high. It would be very interesting to know if there are enzymes with an even lower $K_m$. Very intresting question.