Periodicity of hash functions
What you're actually looking for is called the cycle length most of the time in cryptography.

For a random mapping $f:\{1,2,\ldots,k\} \rightarrow \{1,2,\ldots,k\}$ the cycle length is expected to be $\sqrt{\pi k/8}$ and the tail length is expected to be the same.
If you want more details and some fancy illustrations, the Handbook of Applied Cryptography (PDF) has them.

If you model a cryptographic hash function as a random mapping (which you do most of the time), then you'd expect a cycle length of $\sqrt{\pi 2^n / 8}=\sqrt{\pi}\cdot 2^{\frac{n-3}{2}}$ which is $\approx 2^{127.325748064736}$ for $n=256$, which is the case for SHA-256.

So the TL;DR is:
No it has not been proven that no such $n,k$ exist, but chances are that they exist and range in the area of the values I showed above, which are of no practical relevance as of today.