By a numerical comparison question, we mean the following type of problems :
Given distinct positive real numbers $a$ and $b$, prove by hand without using any computing requirements, that $a > b$ or $a < b$, while the numerical values of both $a$ and $b$ are (at least theoretically) clear.
For example, a post I recently see asks which one of $a = 2^{1000}!$ and $b = 2^{1000!}$ is bigger. For another example, we have people asking which one of $a = \pi^e$ and $b = e^{\pi}$ is bigger. A question comparing $\pi^e$ to $e^\pi$
We emphasize that both $a$ and $b$ should have a theoretically clear value, although sometimes being difficult as the numbers are large or contain well established constants like $\pi$, $e$, $\gamma$. In other words, $a$ and $b$ should not contain symbols where good approximation of that symbol already involves substantial original effort.
When natural constants are involved in this type of question, the solution will most definitely have to dive substantially into the definition of the involved natural constants, making the problem more challenging than the case containing no natural constants.
Question : what are some examples of such numerical comparison questions that does not contain any natural constants (or even only rational numbers and radicals), while having a (challenging) (elementary) solution ?
