What is it called when $x$ grows at a rate of $x^2$?

In your example, if we call the $n^{th}$ term $a_n$, then $a_n$ is a double exponential function (https://en.wikipedia.org/wiki/Double_exponential_function). So my guess is it is called double exponential growth.

When it is a decay instead of growth, it's called quadratic convergence.

This a sequence. So call the first one $a_0$, then the next one $a_1$ and so on. From the ones you listed

$$ a_0=2=2^1=2^{2^0}\\ a_1=4=2^2=2^{2^1}\\ a_2=16=2^4=2^{2^2}\\ a_3=256=2^8=2^{2^3}\\ a_4=65536=2^{16}=2^{2^4}\\ $$

So the pattern is

$$ a_n=2^{2^n}\\ $$

You can prove this by induction. Suppose you know this pattern holds for $a_n$, then the following:

$$ a_{n+1}=a_n^2=2^{2×2^n}=2^{2^{n+1}}\\ $$

shows it holds for $a_{n+1}$ as well. It already worked for $a_0$ through $a_4$, so together that means it holds for all n. So you can see that it grows like the exponential of an exponential. It is super-exponential (careful searching that though since others use different definitions for the same word).