9. The sequence $1,1,2,1,1,3,1,1,1,4,1,1,1,1$, $5, \cdots, \underbrace{1,1, \cdots, 1}_{n-1 \uparrow}, n, \cdots$ has the sum of its first 2007 terms as
9.3898 .

In the sequence, from number 1 to $n$ there are $n+\frac{n(n-1)}{2}$ terms, thus, from number 1 to 62 there are $62+\frac{62 \times(62-1)}{2}$ $=1953$ terms, followed by 54 ones.
Therefore, the sum of the first 2007 terms is
$$
\begin{array}{l}
(1+2+\cdots+62)+(1+2+\cdots+61)+54 \\
=3898
\end{array}
$$