A high-school student named Erna Fekete made a conjecture to me via email three years ago, which I could not answer. I've since lost touch with her. I repeat her interesting conjecture here, in case anyone can provide updated information on it.
Here is how she phrased it. Let $b(0) = 1$ and $b(n)= \tan( b(n-1) )$. In other words, $b(n)$ is the repeated application of $\tan(\;)$ to 1: $$\tan(1) = 1.56, \; \tan(\tan(1)) = 74.7, \; \tan^3(1) = -0.9, \; \ldots $$
Let $a(n) = \lfloor b(n) \rfloor$. Her conjecture is:
Every integer eventually appears in the $a(n)$ sequence.
This sequence is not unknown; it is A000319 in Sloane's integer sequences. Essentially hers is a question about the orbit of 1 under repeated $\tan(\;)$-applications. Her and my investigations at the time led us to believe it was an open problem.