The Heisenberg uncertainty principle states that
$$\Delta x\Delta y\geq\hbar/2$$
Since the magnitude of $\hbar$ is $10^{-34}$ we could measure both $x$ and $p$ with an uncertainty magnitude of $10^{-17}$, but we could also measure $x$ down to an uncertainty magnitude of $10^{-25}$, causing $\Delta p$ to be bounded by $10^{-9}$. For all I know, we could also have $\Delta x$ about $10^{-100}$, at least in principle, thus causing $\Delta p$ to be no better than $10^{66}$, that is, the values the momentum can take are wildly spread out and nearly-completely uninformative.
Now the question: what is the smallest $\Delta x$ (or $\Delta p$) ever achieved in experiments? Is there any theoretical lower bound, except for $0$?