Example 1 Calculate
$$
S\left(9 \times 99 \times 9999 \times \cdots \times \underset{2^{\circ} \uparrow}{99 \cdots 9}\right) \text {. }
$$
(1992, USA Mathematical Olympiad)
[Analysis] If you are familiar with Lemma 1 and have a bit of a sense of magnitude, you can get the answer right away.
Let $N=9 \times 99 \times 9999 \times \cdots \times \underset{2^{\cdots} \cdots 9}{99}$. Then
$$
N<10^{1+2+\cdots+2^{\cdots-1}}<10^{2^{n}}-1 \text {. }
$$

Using Lemma 1, we get
$$
\begin{array}{l}
S(9 \times 99 \times 9999 \times \cdots \times \underbrace{99 \ldots 9}_{2^{n} \uparrow}) \\
=S(N(\underbrace{99.9}_{2^{n} \uparrow}))=9 \times 2^{n} .
\end{array}
$$