When playing in a Python console I observed $\,\pi^\pi\approx 36.46\,$ and $\,e^e\approx 15.15$, and that their ratio is close to $\dfrac{12}{5} =$ a fraction with small numerator and denominator, hence $5\pi^\pi\approx 12\,e^e$.
Do you see a way to solve $$5\,\pi^\pi\;\stackrel{?}{\lessgtr}\; 12\,e^e$$ without calculator use ?
I would estimate that many Math.stackexchangers do not appreciate this kind of "without calculator" questions.
It is asked here in the hope that some elegant solution is found/presented, possibly in the spirit of
this awesome answer
to $\pi^e\stackrel{?}{<} e^\pi$ as of ten years ago.
I have none to offer.
My approach was to consider the natural logarithm of the quotient
$$\log\frac{\pi^\pi}{e^e} = \pi\log\pi - e\log e$$
and to relate it somehow to the definite integral of the logarithm
$$\int_1^t\log x\:dx \;=\; t\,(\log t - 1) +1\,,$$
but without seeing a tangible result.
