As mentioned twice already in the other answers, the number $5$ according to the Von Neumann construction of the natural numbers fits the criteria that you are looking for. It can be seen from the construction that for any natural number $n$ and natural number $m$ with $m<n$ that $m\in n$ as well as $m\subseteq n$. (In fact, with this construction of the natural numbers, this is how the less than relation is defined in the first place!)
A simpler construction which is easier to read which also works is this:
That is, the set contains the empty set enclosed in no brackets, one pair of brackets, two pairs of brackets, on up to four pairs of brackets. Notice that $\emptyset$ is a subset of the above trivially since $\emptyset$ is a subset of every set. Further, the empty set enclosed in a $k$ sets of brackets (with $1\leq k\leq 4$) is a subset because $A$ has as an element the empty set enclosed in $k-1$ sets of brackets as well.