2. Let vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ satisfy $|\boldsymbol{a}|=|\boldsymbol{b}|=1, \boldsymbol{a} \cdot \boldsymbol{b}=\frac{1}{2}$, and the angle between vectors $\boldsymbol{a}-\boldsymbol{c}$ and $\boldsymbol{c}-\boldsymbol{b}$ is $\frac{\pi}{3}$. Then the maximum value of $|\boldsymbol{c}|$ is
2. $\frac{2 \sqrt{3}}{3}$.

Let $\overrightarrow{O A}=a, \overrightarrow{O B}=b, \overrightarrow{O C}=c$. We discuss in two cases.
(1) As shown in Figure 2, points $O, A, C, B$ are concyclic, and point $C$ lies on the minor arc $\overparen{A B}$. At this time, $O C$ is the diameter of the circle, and
$$
|c|=|\overrightarrow{O C}|=\frac{2 \sqrt{3}}{3} \text {. }
$$
(2) As shown in Figure 3, let the reflection of point $O$ across line $A B$ be $O^{\prime}$. Points $O^{\prime}, A, C, B$ are concyclic, and point $C$ lies on the minor arc $\overparen{A B}$. At this time,
$$
\begin{array}{l}
|\boldsymbol{c}|=|\overrightarrow{O C}|<\frac{2 \sqrt{3}}{3} . \\
\text { Therefore, }|\boldsymbol{c}|_{\max }=|\overrightarrow{O C}|_{\max }=\frac{2 \sqrt{3}}{3} .
\end{array}
$$