In UK 40 years ago the calculus was really rigorous first studying limits and series, we did virtually proved all results, except perhaps some integrals but we also had to memorize all formulas as we did not have formula sheet in the exams like they have today in UK A level. In Spain it is still like UK traditional syllabus from back then. I was shocked to ...
This is a bit obvious I think, but when you introduce sequences and their notation in either an algebra or calculus class, you should certainly show students the Collatz Conjecture as one of the examples.
It takes a lot of browsing to find problems somehow related to calculus or analysis, but this is a great MathOverflow list: Not especially famous, long-open problems which anyone can understand. Here are a few from
Are there an infinite number of primes $p$ such that the repeating
part of the decimal expansion of $1/p$ has length $p-1$?