I have seen this question on the internet and was interested to know the answer.

Here it is :

Calculate $\lim\limits_{n\to\infty}(1+\sqrt[2]{2+\sqrt[3]{3+\dotsb+\sqrt[n]n}})$?

Edit : I really tried doing it but wasn't able to get somewhere.

I know how to do questions like $ y = (1+\sqrt{1+\sqrt{1+\dotsb+\sqrt 1}}) $ and then we write $ (y-1)^2 = y $ and solve.

But for this I have no method. So I would like even a sort of a hint to try to get me started, no need for answer.