What is the biggest x such that there is a number n such that x^{n}>n?

I have a rough idea of what the answer is, but I got it by guess-and-check. How do i solve the question (apart from guess-and-check)? I think that x should be somewhere near 1.444667732455312658679247306 but why?

I think e could be the n such that x^{n}-n is greatest.

Please answer if you have an idea of how to solve this.

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