I personally don't ask questions that I think cannot be answered, and advise others not to either.

The questions I do ask, I think have answers, otherwise I wouldn't ask them.

Also, @unenlightened just shared a great quote in another thread of mine that seems relevant here:

For an answer which cannot be expressed the question too cannot be expressed.
The riddle does not exist.
If a question can be put at all, then it can also be answered. — Ludwig Wittgenstein, Tractatus Logico-Philosophicus

I would say that "unanswerable questions", or the incoherent attempt to pose questions to which answers cannot be found, is more the domain of religion (along with positing answers that, they say, cannot be questioned), which I consider half of the exact opposite of philosophy, which I call phobosophy. (The other half is nihilism).

Beyond the simple metaphor of a person exercising their body not intended for hypertrophy, we can exercise our brains rather for overall health and fitness.

At the risk of redundancy we know that physical science relies on our sense of wonderment in discovering and uncovering novel things: all events must have a cause.

Similarly, and perhaps more metaphorical, would be the songwriter who effectively asks the same kinds of questions to him or herself in order to create new music.