Sure, there are lots of things that fit in your category-- there's nothing particularly unique about the case of God (and it's not even one case, as there are many, many different conceptions of God, each of which can be argued separately.)
First of all, any ethical claims are going to be outside the bounds of "proof", because "is" does not imply "ought".
Second, anything that involves interpretation is going to be very difficult to prove, because in order to do so, you need to first agree on an objective standard against which to measure interpretations, and the grounding of this objective standard is open to all kinds of epistemological/ontological problems (the attempted solution of which often leads to onto-theology, bringing us back to the start.)
Third, any problems which are not falsifiable, such as questions regarding eternity, or many cosmological questions (such as what there was prior to the big bang, or what will come after the universe contracts to a point.)
Fourth, any questions regarding why things are, beginning with the classic "Why is there something rather than nothing?"
Fifth, any questions regarding the foundations of logic. (This may be excluded by your mathematical objection, but I'm putting it here just in case.) We cannot use logic to prove that logic is well-founded.
I could go on all day, but in order to cut this short, let's reverse the question: what kind of problems actually are provable? We can start with the notion of falsifiable propositions, but we then have to narrow this down further, as many of these are only falsifiable based upon standards of observation which we may not be able to meet. Further, any proofs are going to be ultimately based upon unproven axioms, and unproven methodological techniques. So, at the end of the day, very little is provable, depending on the strictness of your notion of "proof."
You stipulate that "there is a real world and we can ever better comprehend it", but in doing so, you raise the very question you are claiming to investigate. Just as there are competing arguments for the existence and non-existence of God, none of which are completely probative to an observer unwilling to accept certain axioms, so are there similarly competing arguments for the existence or non-existence of the coffee cup on my desk.
In other words, the fact that the existence of God cannot be conclusively proven or disproven is not an additional claim in support of the non-existence argument. (Sorry to disappoint...)