There is an unambiguous sense in which multiplication is repeated addition, and raising to a power is repeated multiplication. There are even nice maps between them via the log function. I have a strong intuition that exponentiation is in some sense the next step in this series, but efforts to structure the parallels exactly elude me, and I think there are senses in which the series of operations {addition, multiplication, raising to a power} extends naturally to exponentiation and other senses in which it fails to do so.
I would like some better understanding of the ways that the exponential function can and can not be said to be analogous to repeated raising to a power, and why the simple repetition of the preceding function breaks down when it reaches exponentiation. If expressing this question in the complex domain or over other fields helps to clarify it, I would welcome that.
Though my undergraduate days are far behind me, my math level is through calculus of several variables (Apostol) and undergraduate abstract algebra (Pinter).
You can consider an alternative by means of logarithms, generalizing
$$\log ab:=\log a+\log b,$$
or $$ab:=e^{\log a+\log b}.$$
Then iterating the logarithm,
$$\log\log a\text^b:=\log\log a+\log\log b,$$
i.e.
$$a\text^b:=e^{\log a\log b}=a^{\log b}=b^{\log a}.$$
Such an "exponentiation" is associative and commutative and has the neutral element $e$.