11. (20 points) Let $[x]$ denote the greatest integer not exceeding the real number $x$. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=1, a_{n+1}=1+\frac{1}{a_{n}}+\ln a_{n} .
$$

Let $S_{n}=\left[a_{1}\right]+\left[a_{2}\right]+\cdots+\left[a_{n}\right]$. Find $S_{n}$.
11. Combining the known information, from $a_{1}=1$, we get $a_{2}=2$.

Thus, $a_{3}=\frac{3}{2}+\ln 2$.
Since $\frac{1}{2}1$, we have $f^{\prime}(x)>0$, so $f(x)$ is monotonically increasing in the interval $\left(2, \frac{5}{2}\right)$, that is,
$$
\begin{array}{l}
f(2)2, \\
f\left(\frac{5}{2}\right)=1+\frac{2}{5}+\ln \frac{5}{2}<1+\frac{2}{5}+1<\frac{5}{2},
\end{array}
$$

Therefore, $2<a_{k+1}<\frac{5}{2}$, meaning that when $n=k+1$, the inequality holds.
Based on (1) and (2), we know that when $n \geqslant 3\left(n \in \mathbf{Z}_{+}\right)$,
$$
\begin{array}{l}
2<a_{n}<\frac{5}{2} . \\
\text { Hence } S_{n}=\left[a_{1}\right]+\left[a_{2}\right]+\cdots+\left[a_{n}\right] \\
=1+2(n-1)=2 n-1 .
\end{array}
$$