so I encountered in this question and I don't know if im right or not .. that's why i'm here and I want to make sure :). $\mathcal P(\mathbb R)\setminus \{ \{\},\mathbb R\}$ The relation is "subset of" ($\subseteq$).
so there is no Greatest or Least (because the Least was $\{\}$ and the Greatest was $\mathbb R$)). the minimal is all the subsets that contain 1 element inside them (I.e. $\{1\} , \{2\}\ldots$) and the maximum is all the subsets that contain infinite number of elements and can be only subset of the $\mathbb R$.
I don't know how to explain the maximum properly I think. anyways , is it right?
The maximal element is one for which there is no (strictly) "bigger" element, with respect to the relation you are looking at. In your case, those are all sets of the form $\mathbb R\setminus\{a\}$ for $a\in\mathbb R$ i.e. the sets of the form $(-\infty,a)\cup(a,\infty)$.
You have got right the other three properties: the minimal elements are $1$-element sets, and there is no smallest (least) or biggest (greatest) set of all.