Consider the sequence of polynomials $P_1(z)=z$ and
$$P_{n+1}(z)=z+P_n(z)^2.$$
We can think of the Mandelbrot set as the intersection of the images of the disk of radius 2
$$\bigcap_{n=1}^\infty P_n^{-1}\left(\overline{B_2}(0)\right).$$
Each of the terms in the intersection is called a Mandelbrot set lemniscate and they look like smooth approximations to the Mandelbrot set. Now the question is, it doesn't seem intuitive that a set defined like this could be not locally connected. That question is open for the Mandelbrot of course. So is there a sequence of polynomials $\{Q_n\}_{n\in\mathbb N}$ such that $Q_{n+1}^{-1}\left(\overline{\mathbb D}\right)\subset Q_n^{-1}\left(\overline{\mathbb D}\right)$ and the set
$$\bigcap_{n=1}^\infty Q_n^{-1}\left(\overline{\mathbb D}\right)$$
is connected but not locally connected?
The answer is yes.
It is known that for some values of $c$ in the Mandelbrot set, the filled-in Julia set of $f_c(z):=z^2+c$ is not locally connected (this happens for example if $f_c$ has a Cremer fixed point, i.e. there exists a $z$ such that $f_c(z)=z$ and $f_c'(z)=e^{2i\pi \alpha}$ for some $\alpha \in \mathbb R$ with "bad" number theoretic properties).
This map $f_c$ has an escape radius $R>0$, i.e. we have $K(f_c)=\bigcap_{n \geq 0} f_c^{-n}(\mathbb D(0,R))$.