Here is an interesting notion voiced by Fredric M. Menger in Nature Chemistry (DOI): there are questions in chemistry that are impossible to answer. This type of question does not belong to one on the list of unsolved problems in chemistry for example the origin of chirality but ranks among impossible questions found in other branches of science such as the Heisenberg uncertainty principle or Gödel's incompleteness theorems.

Menger sets out to prove his point with an example involving a poly(allylamine). This polymer (DP 1000) can be used as a catalyst in the hydrolysis of certain phosphodiesters. Catalytic activity shows a large variation depending on how the polymer chain is functionalized with carboxylic acids groups with respect to degree and type. Only the average catalytic activity can be measured but it may well be that only 1% or even 0.01% of the polymer chains generate the bulk of it. Addressing all 10¹⁶ individual chains in a given sample is of course not an option.
But is this truly a case of fundamental uncertainty in chemistry? In biochemistry the protein folding problem is all about the prediction of the three-dimensional structures of proteins and enzymes and strategies to tackle this problem. DNA enzymes can be synthesized by sequencing to precise specifications. It is just that compared to biomolecules the particular polyallylamine adds additional complexity.
A more viable impossible question that is briefly touched upon in the Menger article is the uncertainty encountered in molecular dynamics: one can never be certain a given structure is located on the global minimum of the energy surface. In chemistry we can be certain that there is always a possibility that a sinkhole lurks somewhere in the vicinity.