Creative Operation

This is another obvious cheat. $\large \frac{d(2\text{×}2\text{×}2)}{dx}=\frac{d(3^{2^{2^2}}-1)^{2}}{dx}$

A non-cheating is $\large 2×2×2 =\sqrt{(3^{\sqrt{2^{\sqrt{2^2}}}}-1)^{2}}$

A binary operation on $\mathbb{N}$ is any function $f:\mathbb{N}\times \mathbb{N}\to \mathbb{N}$ . So we can define a function $f$ such that $f(2,2)=3$ , $f(3,2)=\left(3^{2^{2^2}}-1\right)^2$ and defined in any way we want for any other pair (For example $f(x,y)=0$ if $x\neq 2,3$ and $y\neq 2$ ), and note it like $f(x,y)=x\otimes y$ .

Then: $(2\otimes 2)\otimes 2 =3\otimes 2=\left(3^{2^{2^2}}-1\right)^2$ And since any operations and functions can be used, $\otimes$ can. And there are infinitely many other ways to do it.

So, if you cannot think of an operation that works... create one yourself.