$\begin{array}{l}\text { 12. } \max _{a, b, c \in \mathbf{R}_{+}} \min \left\{\frac{1}{a}, \frac{1}{b^{2}}, \frac{1}{c^{3}}, a+b^{2}+c^{3}\right\} \\ = \\ .\end{array}$
12. $\sqrt{3}$.

Let $t=\min \left\{\frac{1}{a}, \frac{1}{b^{2}}, \frac{1}{c^{3}}, a+b^{2}+c^{3}\right\}$, then $0<t \leqslant \frac{1}{a}, 0<t \leqslant \frac{1}{b^{2}}, 0<t \leqslant \frac{1}{c^{3}}$.
Thus, $a \leqslant \frac{1}{t}, b^{2} \leqslant \frac{1}{t}, c^{3} \leqslant \frac{1}{t}$.
Therefore, $t \leqslant a+b^{2}+c^{3} \leqslant \frac{3}{t}$.
Hence, $t \leqslant \sqrt{3}$, and when $a=b^{2}=c^{3}=\frac{\sqrt{3}}{3}$, $t=\sqrt{3}$.
Therefore, $\max _{a, b, c \in \mathbb{R}} \min \left\{\frac{1}{a}, \frac{1}{b^{2}}, \frac{1}{c^{3}}, a+b^{2}+c^{3}\right\}=\sqrt{3}$.