What Is Exponentiation? Is there an intuitive definition of exponentiation?
In elementary school, we learned that
$$
a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times})
$$
where $b$ is an integer.
Then later on this was expanded to include rational exponents, so that
$$
a^{\frac{b}{c}} = \sqrt[c]{a^b}
$$
From there we could evaluate decimal exponents like $4^{3.24}$ by first converting to a fraction.
However, even after learning Euler's Identity, I feel as though there is no discussion on what exponentiation really means. The definitions I found are either overly simplistic or unhelpfully complex. Once we stray from the land of rational powers into real powers in general, is there an intuitive definition or explanation of exponentiation?
I am thinking along the lines of, for example, $2^\pi$ or $3^{\sqrt2}$ (or any other irrational power, really). What does this mean? Or, is there no real-world relationship?
To draw a parallel to multiplication:
 If we consider the expression $e\cdot \sqrt5$, I could tell you that this represents the area of a rectangle with side lengths $e$ cm and $\sqrt5$ cm. Or maybe $e \cdot \pi$ is the cost of $\pi$ kg of material that costs $e$ dollars per kg.
Of course these quantities would not be exact, but the underlying intuition does not break down. The idea of repeated addition still holds, just that fractional parts of terms, rather than the entire number, are being added.
So does such an intuition for exponentiation exist? Or is this one of the many things we must accept with proof but not understanding?
This question stems from trying to understand complex exponents including Euler's identity and $2^i$, but I realized that we must first understand reals before moving on the complex numbers.

Let us consider $a^b$. Then, the concept of exponentiation can be understood as the value of $e^{b\ln a}$, since both this expression and $a^b$ are the same.
So, for example, if $b$ is an imaginary number $di$, then $a^b$ denotes a rotation by $d\ln a$ in the complex plane.
However, one can also consider arbitrary exponents $A^B$. This can be interpreted as the (hyper-)volume of a (hyper-)cube in dimension $B$ Euclidean Space $\mathbb{R}^B$ with side length $A$.
How about $2^\pi$, like your example? Assuming you know knowledge about fractals, I believe that $A^C$, with $C\in\overline{\mathbb{Q}}\left(\mbox{ and maybe }C\in\overline{\mathbb{Q}}\cup\mathbb{Q}\right)$ can be interpreted as the  volume of a cube in fractal-dimensional Euclidean space $\mathbb{R}^B$ (if such a thing exists - I don't know) with side length $A$.
As is probably visible, I am a geometer, and I may not have explained what others might know better. However, I have tried my best to explain my intuition of the meaning of $A^B$.
