Any solution to $\zeta(s) = 1$ must have real part $\sigma \le \sigma(1)$ where $\sigma(1)$ is equal to the unique solution $\sigma>1$ of the equation $$\zeta(\sigma)=\frac{2^\sigma+1}{2^\sigma-1}$$ Numerically $\sigma(1)=1.9401016837\dots$
$\sigma(1)$ is the best possible constant here.
See the paper arXiv:1107.5134 where other problems of this type are considered.
About the solutions of $\zeta(\zeta(s))=1$. There are a countable set of solutions to
$\zeta(s) = 1$. $s_1$, $s_2$, $s_3$, $\dots$ These solutions are situated mainly very near the
critical line. Near the pole $\zeta (s)\sim (s-1)^{-1}+ \gamma $
if you solve $(s-1)^{-1}+\gamma = 1/2+it$ you find a circle with center at
$1+(1/2-\gamma)^{-1} /2=-5.47537$ and radius 6.47537.
Therefore one expects to find solutions of $\zeta(s) = s_k$ for points simultaneously
near this circle and near the point 1.
With Mathematica you find for example one of this in this way
a = s /. FindRoot[Zeta[s] - 1, {s, N[ZetaZero[10000]]}, WorkingPrecision -> 50]; FindRoot[Zeta[s] - a, {s, 1 + (a - EulerGamma)^(-1)}, WorkingPrecision -> 50]
That gives you the solution
s= 1.0000000002505104088470167417362938109319852591130 - 0.00010123550290056930653742177989540980110048808885363 I
That satisfies $\zeta(\zeta(s))=1$