What is the purpose of multiple key addition layers in AES?
Suppose you remove the key addition layer. What happens then?

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   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.