3. Given that $x_{1}, x_{2}, \cdots, x_{2010}$ are all positive real numbers. Then
$$
x_{1}+\frac{x_{2}}{x_{1}}+\frac{x_{3}}{x_{1} x_{2}}+\cdots+\frac{x_{2010}}{x_{1} x_{2} \cdots x_{200}}+\frac{4}{x_{1} x_{2} \cdots x_{2010}}
$$

the minimum value is $\qquad$
3.4.

Starting from the last two terms, repeatedly applying the AM-GM inequality, we get
$$
\begin{array}{l}
\text { Original expression }=\sum_{i=1}^{2010} \frac{x_{i}}{\prod_{j=1}^{i-1} x_{j}}+\frac{4}{\prod_{j=1}^{2010} x_{j}} \\
=\sum_{i=1}^{2009} \frac{x_{i}}{\prod_{j=1}^{i-1} x_{j}}+\left(\frac{x_{2010}}{\prod_{j=1}^{2009} x_{j}}+\frac{4}{x_{2010} \prod_{j=1}^{2009} x_{j}}\right) \\
\geqslant \sum_{i=1}^{2009} \frac{x_{i}}{\prod_{j=1}^{i-1} x_{j}}+\frac{4}{\prod_{j=1}^{2009} x_{j}} \geqslant \cdots \geqslant x_{1}+\frac{4}{x_{1}} \geqslant 4,
\end{array}
$$

where the equality holds if and only if
$$
x_{2010}=x_{2009}=\cdots=x_{1}=2 \text {. }
$$

Therefore, the minimum value sought is 4.