A non-empty subset $S ⊆ \mathbb{R}$ is bounded above by $k ∈ \mathbb{R}$ if $s ≤ k$ for all $s ∈ S$. The number $k$ is called an upper bound for $S$.
Could by this definition we say that $S$ can have the greatest element and that element is the upper bound?
By that definition, if $S$ has a greatest element, then that element is an upper bound of $S$. That's the only case in which an element of $S$ is also an upper bound of $S$.
On the other hand, even if $S$ has no greatest element, it still can have upper bounds.