limit of the sequence $x_{n}:= \sqrt[n]{n \sqrt[n]{n \sqrt[n]{n\ldots}}}$

Yes, the limit is $1$: $$x_n=n^{\sum_{k=1}^{\infty}\frac{1}{n^k}}= n^{\frac{1}{n-1}}=e^{\frac{\ln(n)}{n-1}}\to 1.$$