Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}$

I'm so puzzled about this:

$$a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}.$$

Why isn't $a^{b^c}$ equal to $a^{(bc)}$? Why is $a^{b^c}$ instead equal to $a^{(b^c)}$? And how is it possible that $(a^b)^c = a^{(bc)}$?

My mind is pretty much exploding from trying to understand this.

That $a^{b^c}$ stands for $a^{(b^c)}$ rather than for $(a^b)^c$ is merely a notational convention; we say that the exponentiation notation associates to the right (whereas arithmetic operations associate to the left, so that $a-b-c$ means $(a-b)-c$ rather than $a-(b-c)$).

The fact that $(a^b)^c=a^{b\times c}$ is easy to understand: $a^b$ is obtained by multiplying together a sequence of $b$ copies of $a$, and $(a^b)^c$ is obtained by multiplying together $c$ such products; writing all this out in terms of copies of $a$ means that $b\times c$ such copies have been multiplied together. Now you can see also why this is not equal to $a^{(b^c)}$ which is obtained by multiplying together $b^c$ copies of $a$.

Finally the fact that $(a^b)^c=a^{b\times c}$ explains why the convention is that exponentiation notation associates to the right: both $a^{(b^c)}$ and $(a^b)^c$ are useful expressions, but since the latter can be more easily written as $a^{bc}$, one might as well reserve $a^{b^c}$ to stand for the former, which has no such easy alternative. It might seem that $a^{b^c}$ is such an enormous product that is unlikely to be useful; it is however encountered surprisingly often in some contexts.

Note that $(2^2)^3=4^3=64$, whereas $2^{(2^3)}=2^8=256$ (and incidentally $2^{2\cdot 3}=2^6=64$).