Hitting Set Problem with non-minimal Greedy Algorithm
The greedy algorithm always finds a minimal set (inclusion-wise). However, it does not necessarily find a minimum one.

So what does minimal and minimum mean, you might ask. Minimal set in a family of sets (say the set of all hitting sets), is a set that is not a superset of any other set in the family. That means, a minimal hitting set is a hitting set where if you exclude any element from it becomes a non-hitting set. On the other hand, a minimum hitting set is a hitting set of the smallest size. Even though a minimum hitting set might not be unique, all minimum hitting sets are minimal but minimal hitting sets do not have to be minimum.

An example, where the greedy algorithm does not yield a minimum hitting set is

$$U = \{1, 2, 3, 4, 5\}\\ 
\mathcal{F} = \{
\{1, 2, 3\}, \{1, 3, 4\}, \{1, 4, 5\}, \{1, 2, 5\}, \{2, 3\}, \{4, 5\}
\},$$
where $U$ is the universe and $\mathcal{F}$ is the family of sets to be hit. A minimum solution is hence $\{2, 4\}$. Any solution including $1$ is not minimum, since the set must hit $\{2, 3\}$ and $\{4, 5\}$ using at least one element of each resulting in a solution of size at least 3. However, the greedy algorithm includes $1$ in the first step and hence can not yield a minimum solution.