A tower of irrationals? Set $x_0:=1$ and for every $n\in\mathbb{N}$ set $x_n:=2^{\frac{1}{2}x_{n-1}}$, so
$$x_1=\sqrt{2},\; x_2=\sqrt{2}^{\sqrt{2}}, x_3=\sqrt{2}^{\sqrt{2}^{\sqrt{2}}},\ldots$$
Undoubtedly the $x_n$ are irrational for all $n\in\mathbb{N}=\{1, 2,\ldots\}$. (How) can we prove that?


Undoubtedly the $x_n$ are irrational for all $n∈N={1,2,…}$

The proof of that statement would earn you a fair amount of fame, since the irrationality of $x_n$ for $n>2$ is still an open problem. 
Indeed, one can use the Gelfond-Schneider theorem to prove that $\sqrt 2^{\sqrt 2}$ is transcendental:

Theorem (Gelfond-Schneider). If $a$ and $b$ are algebraic numbers with $a ≠ 0, a ≠ 1$, and $b$ irrational, then any value of $a^b$ is a
  transcendental number.

However, currently we have no tools to assess the irrationality of numbers such as $\sqrt 2^{{\sqrt 2}^{\sqrt 2}}$ where the exponent is transcendental. 
There exist however other related numbers whose rationality is known, for instance the infinite tower
$$
\sqrt 2^{{{\sqrt 2}^{{\sqrt 2}^{\ldots}}}} = \sqrt[\leftroot{-2}\uproot{2}4]{4}^{{\sqrt[\leftroot{-2}\uproot{2}4]{4}}^\ldots} = 2,
$$
which is the value of $\displaystyle \lim_{n\to\infty} x_n$. In particular, we have the following lemma.

Lemma. Let $n$ be a positive integer different than $1$, $2$ and $4$. Then the infinite power tower 
  $$\sqrt[\leftroot{-2}\uproot{2}n]{n}^{{\sqrt[\leftroot{-2}\uproot{2}n]{n}}^\ldots}$$
  is transcendental.

In particular, 
$$\sqrt[\leftroot{-2}\uproot{2}3]{3}^{{\sqrt[\leftroot{-2}\uproot{2}3]{3}}^\ldots}$$
is a transcendental number.

Finally, I give the following interesting reference on infinite power towers:
Mladen V-M, Some Results On Infinite Power Towers, Notes on Number Theory and Discrete Mathematics (2010), 3, 18-24. 
It is freely accessible at:
http://www.nntdm.net/papers/nntdm-16/NNTDM-16-3-18-24.pdf
