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.

$a^b$ refers to the "multiplicative power" of performing b multiplications by a. This is intuitively obvious with positive integer 'b's, but still holds for fractional and negative values when you put a little brain grease into considering what it means to do 'half a multiplication' or a 'negative multiplication'.
$9^\frac{1}{2}$ is 3 since multiplying by 3 is half the multiplication that multiplying by 9 is. (eg, it's the multiplication which, if done twice, will be equivalent to multiplying by 9). Similarly for $8^\frac{1}{3} = 2$, etc.
$4^{-1}$ is 1/4, since multiplying by 1/4 is 'unmultiplying' once by 4. (eg, it's the operation that gets cancelled out by multiplying once by 4.)
$a^0$ (including $0^0$) is 1, since doing no multiplication to something is the same as multiplying that something by 1.
