For example, let $X=\left(X_t\right)_{t\ge0}$ a right-continuous random process, $x\in\mathbb{R}$ and $T_x = \inf\left\{t\ge0\;;\; X_t\ge x\right\}$.
The random variable $T_x$ is a stopping time that is also a "reaching time". What would look like a stopping time that is not built in such a way ?