Impossibility proofs
Patrick Honner over at Quanta Magazine has an essay on what impossibility proofs in mathematics can teach us. He brings up a number of beneficial effects of exploring such proofs, described here with my interpretation.

At some point in our mathematical training, we start encountering problems that simply don't have a solution. But, what we can do is prove that a solution does not exist. This can be done by showing that the various conditions imposed by the problems cannot be simultaneously satisfied. We learn to always check for every future problem, whether the conditions even admit a solution. And if we build this habit, we have leveled up as a mathematician.

"Proving that something is impossible is a powerful act of mathematics. It shifts our perspective from that of rule follower to that of rule enforcer." This is a even more advanced version of the previous point. The maturity to become a rule enforcer is another leveling up act. In the journey to becoming an intellectual, this is when we start to build intellectual honesty.

One of the goal in mathematics is to understand the consequences of a set of axioms. We realize that this means figuring out the true statements or theorems that can be derived from these axioms, but we must also list out important false statements within this system of axioms. In this way, we determine the boundary of our mathematical system, that discriminates between the true and the false.

Finally, once we have conjectured some statement as false, we might need to create new mathematics, sometimes new fields, to prove that conjecture. This is the domain of the expert mathematician.
As a physicist, I have always been interested in the impossibility results within physics. These have different character than those in mathematics. For instance, the impossibility of going faster than the speed of light in flat spacetime, is axiomatic in relativity. In quantum information, there are a wealth of impossibility theorems, such as the nocloning theorem, the monogamy of entanglement, etc. These place strong constraints on the type of universes that can exist, and which cannot. These theorems often take a central place in the philosophy of physics, and are often understood to characterize a theory as much as its axioms. In fact, one the favorite exercises for bored researchers is to figure out if various theories of physics can be derived from such results taken as axioms.
Perhaps, one day I will write a review article, which will discuss the many major and minor impossibility theorems in physics.