Any branch of mathematics after the first few definitions will make everyone routinely ask themselves some basic questions. I consider Inverse Galois Problem is one such. If the question (i) is not highly technical, (ii) can be understood at very early stages and (iii) does not sound concocted then it justifies itself.
These are the natural questions the subject should attempt to answer. (It is irrelevant if solving them requires Fields medallists or undergraduates).
Let me list more questions in the same category (not necessarily of the same level of difficulty!)
Which divisors of $|G|$ are orders of subgroups of $G$?
Which connected open subsets of the complex plane are biholomorphic to the unit disc?
For which numbers $d$, is the ring $\mathbf{Z}[\sqrt d]$ a UFD?
Which finite groups occur as subgroups of $\mathbf{SO}(3)$?
Which integers are represented by an indefinite/definite integral quadratic form?
Which projective curves are subvarieties of the projective plane?
I have been under the impression that this is the way mathematicians think. If someone questions the relevance of the above questions it would be difficult for me to communicate with that person.
