IMC 2023 Problem 10

For every positive integer $n$ , let $f(n)$ , $g(n)$ be the minimal positive integers such that $$1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}=\frac{f(n)}{g(n)}.$$ Determine whether there exists a positive integer $n$ for which $g(n)>n^{0.999n}$ .