3. For the irrational number $\mathrm{e}=2.71828182845 \cdots$, define the function $f(n)=k$, with the domain being positive integers, where $k$ is the digit at the $n$-th position after the decimal point of $\mathrm{e}$, and it is specified that $f(0)=2$, then the range of $f(f(f(n)))$ is $(\quad)$.
(A) $\{3,4,6,9\}$
(B) $\{2,0,1,9\}$
(C) $\{0,3,5,9\}$
(D) $\{1,2,7,8\}$
3. D.

It is easy to know that the range of $f(n)$ is $\{0,1, \cdots, 9\}$, the range of $f(f(n))$ is $\{1,2,7,8\}$.
Then the range of $f(f(f(n)))$ is $\{1,2,7,8\}$.