There is no way to do this non-analytically. Define the above as a function f(x) = e^x - x^e. We need to prove f(π) > 0, and knowing f(e) = 0, we simply to need to see if the derivative is positive for x > e. An alternative solution is taking the natural log of both sides, and then finding that π/e > ln(π), by approximating ln(π) with a taylor series