Let $A$ be the set of non-intersecting "figure eights" (objects shaped like $8$) in the $\mathbb{R} \times \mathbb{R}$ plane, so that $|A|$ is maximal. What's $|A|$?
Examples of figure eights: $8$, $\infty$, and all angle variations.
It seems to me the answer is $\aleph$. Maybe you can view figure eights as objects defined by center and radius. Center being point of intersection of $8$, and radius defines the two "circles" above and below the eight.
Honestly I'm quite clueless.
For each $8$, we can pick a pair of points with rational coordinates, one in each eye. This gives us an injection (why?) from $A$ to $\mathbb Q^4$, hence $|A|\le\aleph_0$.
Regarding the "why?":
An eight consists of two simple closed curves $C_1, C_2$ that have one point $S$ in common and apart from that each is in the exterior region of the other (recall that by the Jordan curve theorem, a simple closed curve defines a interior (bounded) and an exterior (unbounded) region). So an eight defines three regions: $U_1$ the interior of $C_1$, $U_2$ the interior of $C_2$, and $U_3$ the exterior.
Assume that we have another eight (with $C_1', C_2', S',U_1', U_2',U_3'$) and that the two eights do not intersect. And assume that $a$ is an interior point of both $C_1$ and $C_1'$ and $b$ is an interior point of both $C_2$ and $C_2'$. $C_1$ and $C_1'$ can be in three possible situations: "Winding around each other", i.e. either $C_1\subseteq U_1'$ or $C_1'\subseteq U_1$, or with nonoverlapping interior. However, from $a\in U_1\cap U_1'$ we see that the interiors overlap, hence wlog. $C_1\subseteq U_1'$. Since $C_1\cup C_2$ is connected, also $C_2\subseteq U_1'$ and hence $b\in U_2\subseteq U_1'$. But then $b\notin U_2'$, contradiction.
