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How to show $e^{e^{e^{79}}}$ is not an integer

Is ${^5\pi}$ an integer? It is "obviously" not, right? But can we prove it?

Here ${^5\pi}$ means the result of tetration $\underbrace{\pi^{\pi^{\pi^{\pi^\pi}}}}_{5 \text{ times}}$.

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    @GarouDan "IntegerQ[expr] returns False unless expr is manifestly an integer (i.e. has head Integer)." http://reference.wolfram.com/mathematica/ref/IntegerQ.html2011-12-14

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