A link to the original question for reference:Click here
I tried to study a more general situation:
Let $y_{d,d}=1$ and $$ y_{n+1,d}=\frac{1}{n+1}\left(ny_{n,d}+\left(1+y_{n,d}-y_{n,d-1}\right)^{-n}\right), $$ where $n,d\in\mathbb{N}$ with $n\ge d$.
Using the arguments from the original problem (cf. the link above) (with some modifications), we have $$ y_{n,d}<dy_{n,1}\;\;\mbox{for all}\;\;n\ge d. $$ This imples that $y_{n,d}<\frac{d(Log(n)+1)}{n}$ for all $n\ge d$.
I made an initial attempt (induction) and guessed the lower bound, i.e. $$ y_{n,d}>\frac{d(H_n-H_d)}{n}\;\mbox{for all}\;n\ge d, $$ where $H_n$ is the harmonic series ($H_n=\sum_{j=1}^n\frac{1}{j}$).
Although I can't prove this now, the numerical results really supports this conjecture. Furthermore, I conjectured that $$ y_{n,d}\sim\frac{dLog(n)}{n}\;\mbox{as}\;n\to\infty. $$
Any help will be greatly appreciated.