Fekete's conjecture on repeated applications of the tangent function 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.
 A: This isn't a proof, but's too long for a comment, and may just be a restatement of the problem. 
For contradiction, let $k$ be any integer such that $b(n) = k$ never holds. This means $k \leq a(n) < k+1$ never holds. 
Since $a(n)$ can't be between $k$ and $k+1$, $\arctan a(n)$ can't be either. 
Thus, there's an interval between $-\pi/2$ and $\pi/2$ that a(n) may not touch. Let's call it $[c,d)$. 
Since tan is periodic, $a(n)$ must also avoid $m\pi+[c,d)$.
Since $\pi$ is irrational, $m\pi+[c,d)$ must contain an infinite number of integers (pretty sure this is true, but I could be wrong). 
Therefore, there are an infinite number of intervals (approaching $\pi/2$) that $a(n)$ must avoid. Further, $a(n)$ must avoid the arctans of these intervals, and the arctans of those intervals, etc. The repeated arctan intervals approach 0. 
Of course, $a(n)$ also has to avoid those intervals plus any multiple of $\pi$.
This non-proof actually applies to any interval $a(n)$ misses, so, if true, shows that $a(n)$ is dense in $\mathbb{R}$. Hope that helps.  
A: I had made the same conjecture as Fekete, apparently around the same time -- mid-2007.  In 2008 I verified that the first twenty million terms do not include 319.  (I actually pushed the verification further, but I can't find the more recent records at the moment.)
Because $\tan(x) - x = x^3/3 + O(x^5)$, the function spends a lot of its time in a small neighborhood around $0$.  It escapes when it nears $\pi/2$ and quickly returns for many iterations.
A mostly-unexplained phenomenon presumably related to the above: there are long spans of small numbers followed by short, 'productive' spans with large numbers.  $\tan^k(1)$ is "below 20 or so" (according to a 2008 email I sent) for $360110\le k\le1392490$ but in the next 2000 numbers there are five which are above 20.
More theory is needed!
