Can $\sqrt{p}^{\sqrt{p}^{\sqrt{p}}}$ be an integer, if $p$ is a non-square positive integer? Can $\sqrt{p}^{\sqrt{p}^{\sqrt{p}}}$ be an integer, when $p$ is a non-square positive integer? 
Of course, it seems it would never but is there a proof of the fact, or maybe we have some spooky $p$ that makes it valid?
 A: Although such a number almost surely can not be an integer, it seems that this should be an open problem still.
However the claim that it can not be an integer would follow from Schanuel's conjecture:

Given any $n$ complex numbers $z_1,\dots,z_n$ which are linearly
  independent over $\mathbb{Q}$, the extension field
  $\mathbb{Q}(z_1,\dots,z_n, e^{z_1},\dots,e^{z_n})$ has transcendence
  degree at least $n$ over $\mathbb{Q}$, i.e. the set $\{z_1,\dots,z_n,
 e^{z_1},\dots,e^{z_n}\}$ contains at least $n$ algebraically
  independent numbers.

This is a strong statement.  For example, let $z_1 = 1$ and $z_2 = i\pi$.  Then the set $\{1, i\pi, e, -1\}$ must contain two algebraically independent numbers.  Then $\pi$ and $e$ would be algebraically independent, showing that, for example, $e+\pi$ and $e\pi$ are transcendental (which otherwise is an open problem).
Now consider an algebraic number $\alpha$ which is irrational (such as $\sqrt{m}$ for $m$ not a square).  By the Gelfond-Schneider theorem, $\alpha^\alpha$ is transcendental.  Therefore, $\log \alpha$, $\alpha \log \alpha$, and $\alpha^\alpha \log \alpha$ are $\mathbb{Q}$-linearly independent.
By Schanuel's conjecture, the set $$\log \alpha, \alpha \log \alpha, \alpha^\alpha \log \alpha, \alpha, \alpha^\alpha, \alpha^{\alpha^\alpha}$$
contains at least three elements that are algebraically independent.  But $\alpha$ is algebraic.  Also $\alpha \log \alpha$ and $\alpha^\alpha \log \alpha$ are algebraically dependent on $\log \alpha$ and $\alpha^\alpha$, so $\log \alpha$, $\alpha^\alpha$ and $\alpha^{\alpha^\alpha}$ must be algebraically independent.
In particular,  $\alpha^{\alpha^\alpha}$ must be transcendental.
For more on this subject, including more and stronger consequences on Schanuel's conjecture, please see the papers by Marques and Sondow, Schanuel's conjecture and algebraic powers $z^w$ and $w^z$ with $z$ and $w$ transcendental and The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture.
I would be interested if anyone thought that, with this kind of problem, that proving something is not rational or not an integer might be substantially more tractable than proving it is transcendental.
