Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.
Here’s a question I once received (though I probably changed the exact wording somewhat):
Exponentiation is to multiplication as multiplication is to addition. In other words,
,
,
where the operation is repeated
times.
So, multiplication is to addition as addition is to what?
My kneejerk answer was that there was no answer… while exponents can be thought of as repeated multiplication and multiplication can be thought of as repeated addition, addition can’t be thought of as some other thing being repeated. But it took me a few minutes before I could develop of proof that could be understood by my bright young questioner.
Suppose 
and
However, we know full well that
Therefore, there can’t be an operation analogous to addition as addition is to multiplication or as multiplication is to exponentiation.
Mean Green Math 
What about repeated incrementation a la the Turning machine?
Also, wouldn’t x+0 = x (ie, using the identity) make sense here?
Certainly repeated incrementation would work, and I suggested that to my student at first. However, my student didn’t like that answer, as she didn’t think repeated incrementation (which always starts on 1) was analogous to repeated addition (which starts on
and not 
). Instead, she was looking for a single binary operation combining both 
and 
.