There is no such continuum. See 

Z. Waraszkiewicz, *[Sur un problème de M.H. Hahn](https://eudml.org/doc/212685)*, Fund. Math. 22 (1934) 180–205.

Waraszkiewicz constructed an uncountable family $W$ of continua in the plane called Waraszkiewicz spirals so that no continuum can be mapped continuously onto every continuum in $W$. Start with the space $X=S^1\cup [1,2]$ and consider the maps $f,g:X\to X$ where 

 - $f(x)=x$ if $x\in S^1$, $f(x)=\exp(4\pi x)$ for $1\leq x\leq \frac{3}{2}$, and $f(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$
 - $g(x)=x$ if $x\in S^1$, $g(x)=\exp(-4\pi x)$ for $1\leq x\leq \frac{3}{2}$, and $g(x)=2(x-1)$ for $\frac{3}{2}\leq x\leq 2$

Now $W$ is the collection of all inverse limits of all inverse sequences $(X_i,h_i)$ where $X_i=X$ for all $i$ and $h_i\in\{f,g\}$.

For a short and readable description (where I found the construction) see:

W.T. Ingram, *[Concerning images of continua](http://topology.auburn.edu/tp/reprints/v16/tp16010.pdf)*, Topology Proceedings 16 (1991) 89-93.

This non-existence results has been improved upon a few times, for instance in the following:

S.B. Nadler, *[The nonexistence of almost continuous surjections between certain continua](https://doi.org/10.1016/j.topol.2005.01.042)*, Topology and its Applications, 154 (2007) 1008-1014.