TITLE: In calculus, which questions can the naive ask that the learned cannot answer?
QUESTION [110 upvotes]: Number theory is known to be a field in which many questions that can be understood by secondary-school pupils have defied the most formidable mathematicians' attempts to answer them.
Calculus is not known to be such a field, as far as I know. (For now, let's just assume this means the basic topics included in the staid and stagnant conventional first-year calculus course.)
What are

the most prominent and 
the most readily comprehensible

questions that can be understood by those who know the concepts taught in first-year calculus and whose solutions are unknown?
I'm not looking for problems that people who know only first-year calculus can solve, but only for questions that they can understand.  It would be acceptable to include questions that can be understood only in a somewhat less than logically rigorous way by students at that level.

REPLY [24 votes]: Some discussion of the three-body problem, especially if your students have followed a physics course where Newtonian mechanics have been taught, can be relevant:

It is very easy to state (it is a simple ODE)
It is useful (think space travel)
Very few things are known about it, if we compare against the two-body problem: no closed form of trajectories (series exist though), new periodic solutions have been discovered in 2012, boundedness of solutions is difficult to assess, etc.

As a bonus, this can be used as a motivation for the need of asymptotic expansions.