In Brainiac: Science Abuse, there is a wonderful category called How Hard is Your Thing? (see [1]). Thaila Zucchi makes seemingly hard things undergo some of the ultimate stress tests: Can they stand the heat of thermite? The abrasion of an angle grinder? The impact of a ton of bricks?
There were several occasions in the history of mathematics when mathematicians had to answer to the question: how hard is your maths?
There was a time back at the university, when pretty much any question about mathematics I came up with led to the axiom of choice. This seemingly innocuous and at first sight quite reasonable axiom leads to pretty strange conclusions such as the Banach–Tarski paradox. Roughly speaking, it requires that for any collection of non-empty sets it should be possible to construct an element of the Cartesian product.
Here are two of the questions I had at the time:
Glimpses of a fascinating world 