Exponential iterates of a complex number Let $f:\mathbb C\to \mathbb C$ be defined by $f(z)=e^z-1$.  Let $f^n$ denote the $n$-fold composition of $f$.  
In my new paper Erdős space in Julia sets I show that $$Z:=\{z\in \mathbb C:\lvert\operatorname{Im}(f^n(z))\rvert\to\infty\}$$ contains a homeomorphic copy of the set of points in Hilbert space $\ell^2$ which have all rational coordinates. But I do not know of a specific complex number $z$ which belongs to this set. It is easy to find $z$'s for which the real part goes to infinity;  $\operatorname{Re}(f^n(1+0i))\to\infty$ as $n\to\infty$,  but the imaginary part of $f^n(1+0i)$ is always $0$. So the question is, can you give the precise coordinates of a point in the complex plane which belongs to $Z$? 
How about an  answer to this question for $f(z)=e^z$?
 A: It depends on what you mean by precise coordinates. I am not sure that I would expect to find a number that has a specific closed form. But then, I do not know how to find a "precise" point where the real part tends to infinity under iteration either, had you not chosen a real parameter.
To find such a point up to arbitrary precision, on the other hand, is quite easy. For instance, let $N\geq 0$, define
$$ z_{N,N} := \log N + 2\pi i N. $$
and inductively let $z_{N,j}$ (for $j<N$) be the preimage of $z_{N,j+1}$ in the strip at imaginary parts between $(2j-1)\pi$ and $(2j+1)\pi$.
Then $z_{N,0}$ converges to a point $z_0$ such that $f^j(z_0)$ has imaginary parts between $(2j-1)\pi$ and $(2j+1)\pi$, and in particular the imaginary parts converge to infinity.
Here is some simple python code:
>>> n=100
>>> orbit = [0]*(n+1)
>>> orbit[n] = math.log(n) + 2*math.pi*1j*n
>>> for j in range(n):
          orbit[n-j-1] = cmath.log(orbit[n-j]+1) + (n-j-1)*2*math.pi*1j

I get approximately $z_0 = 2.1302059107690132+1.1190548923421213j$. Of course, given the very strong expansion and instability, if you iterate forward this will only follow the desired orbit for a small number of iterations (for me, it stays close for 10 iterations), but Gottfried Helms has given a higher-precision estimate below.
EDIT. My notation was somewhat messed up above in the original post; it should be better now I hope. Note that the $\log N$ can be omitted in the definition of $z_{N,N}$, and you will converge to the same value.
