What is the purpose of multiple key addition layers in AES? I understand the purpose of the SubBytes layer (confusion) and the ShiftRows and MixColumns layers (diffusion).

But I don't see the purpose of having more than one key addition layer. What I've been able to find is that it can mitigate side channel attacks if the AES implementation leaks information, but I don't see the connection there.
Suppose you remove the key addition layer. What happens then?

(Mouse over to see the answer. Try figuring out yourself first!)


   Denote AES without the key addition layer by $\operatorname{f}$. Suppose an adversary has $\operatorname{f}(x)$. They can recover $x$ by applying $\operatorname{f}^{-1}(x)$. If there is no secret information (the key), then anyone can perform the permutation/inverse permutation at will. You cannot have confidentiality without a secret decryption key.


Edit in response to modified question

If the key is only applied once, then given a ciphertext an adversary can apply the inverse permutation on the ciphertext until they end up with $m' \oplus k$. This allows them to recover $k$ via exclusive-or with $m'$, where $m'$ is the result of $f'(x)$ and $f'$ is the first steps of AES up to the key addition layer.