Well, I've proven that : $$x^{\frac1x}<1.5$$ for $x \in \mathbb R^+$, or more specifically I've shown that the maximum value of the expression is $1.44...$ for $x = e$. But I used calculus (finding the derivative, then finding the critical point and proving it is a local max).
But I'm looking for another method that does not require calculus (induction if possible, that's why I added $x \in \mathbb N$). The induction step seems pretty hard: I have no way to simplify $(x+1)^{\frac1{x+1}}$. Any help would be appreciated. (AM-GM inequality seems a good idea, though I am not able to proceed further..., or more specifically, it's not working for smaller values of x.)
For $x\in\{1,2,3\}$ it's obvious.
For $x\in\mathbb N$ and $x\geq3$ it's enough to prove that $$x^{\frac{1}{x}}>(x+1)^{\frac{1}{x+1}}$$ or $$x^{x+1}>(x+1)^x,$$ for which we can use induction. See here: Prove by induction that $(k + 2)^{k + 1} \leq (k+1)^{k +2}$