What are some ways of thinking about exponentiation, other than repeated multiplication? For instance, we can think of exponentials as solutions to differential equations. Are there any other interesting ways of describing exponents without iteration of multiplication?
 A: Using the exponential function (with different characterisations) $e^x$ and the natural logarithm:
For every $a, b \in \mathbb R$, we can write $a = e^{ln(a)}$ (suppose for now $a>0$) which implies
$$a^b = (e^{ln(a)})^b = e^{b\cdot ln(a)}$$
See these great answers for more detail.
In fact, this allows to expand the concept of exponentials to the complex plane: 
Given $z, w \in \mathbb C$, we define 
$z^w = e^{w\cdot ln(z)}$ (where $ln(z)$ is the complex logarithm), and use the fact that for $w = a + i\cdot b$:
$$e^w = e^{a+ib}=e^a\cdot e^{ib} = e^a(cos(b)+i\cdot sin(b))$$
since $exp(x+y)= exp(x)\cdot exp(y)$ and $e^{ib} = cos(b)+i\cdot sin(b)$. 
However, note that this is a multi-valued function unless a branch of the complex logarithm is chosen (otherwise $(e^{ln(z)})^w \neq e^{w\cdot ln(z)}$).
A: If a function's growth rate is proportional to its value then it's an exponential function.
i.e.  $b^x$ is growing at the rate $c_b*b^x$ for some constant $c_b$.  $b^x$ is the one function where that can be true.
Which allows for the "how can you multiply something by itself 1/2 a time to get $b^{1/2}$" question to be "how much do you have at 1/2 if the instantaneous grate of growth always proportional to the current ammount?"  Of course, that's a pretty hard question to answer without circular reasoning, but it does make non-integer and even non rational exponents meaningful.
A: From the category theory point of view, you can adress a lot of exponentials.
For example, in the category of Sets, an exponent of two sets is the set of functions from one to another 

( A hint to see why it makes sense: how many functions exists from a set $A$ to a set $B$?
  Answer:  

 $|B|^{|A|}$ where $|X|$ is the cardnality of a set $X$

)

https://en.wikipedia.org/wiki/Exponential_object
A: If you go one step beyond multiplication...
$$a^b=\underbrace{a\times a\times a\times\cdots\times a}_b$$
$$a\times b=\underbrace{a+a+a+\dots+a}_b$$
And combine these to turn exponentiation into addition.
$$\color{white}{\text{Way too many additions, so I left it out.}}$$
Take these farther steps, and you'll create the hyperoperation sequence.
