Assume we have a set $L$ of $n$ Jordan curves which intersect each other and divide a plane into regions. Given a region $r$, we characterize it by $C(r)=\{l \in L: \text{$r$ lies inside $l$}\}$. Let us say that regions $r$ and $s$ are equal if they have the same characteristics, i.e. $C(r)=C(s)$. Is it always possible to draw the curves and get $2^n$ regions without equal regions?
The folowing picture demonstrates all the main possibilities for $n=3$ (the third case satisfies our conditions): https://i.stack.imgur.com/NbN4M.jpg
If I understood correctly, you are looking for Venn diagrams (that are Euler diagrams in which all combination of intersections are present). Wikipedia describes the following sequence with this property:
And another one, called Edwards–Venn diagrams:
And finally, an idea of my own that is easily described with explicit formulas: you could choose the regions under the curves
$$2^{-n}\sin(2^n x), \quad x\in[0,2\pi],$$
appropriately cropped to give you finite regions that are bounded by Jordan curves.