Just bash through the 12 cases

Note that if $a, b$ or $c$ is $1$ , then all indices above $1$ wouldn't contribute anything. Therefore, to maximise, we will make $d=1$ .

Now we have to maximise $a^{b^c}$ from $2, 2, 3$ . Note that there are only 3 ways of evaluating this: $3^{2^2} = 81$ , $2^{3^2} = 512$ , $2^{2^3} = 256$ . Therefore, $\boxed{512}$ is the maximum value, achieved at $a=c=2$ , $b=3$ and $d=1$ .

$\text{max}(a^{b^{c^{d}}})$ if $d = 1,$ and $\text{max}(b^c)$ . So $\text{max}(b^c)=3^2=9$ . $\text{max}(a^{b^{c^{d}}})= 2^{3^{2^{1}}} = 2^9 = 512 \\ \text{max}(a^{b^{c^{d}}}) = 512$