Any evidence that one way functions exist?
AES is widely believed to be secure: many smart people have tried to break it and haven't been able.  This is not a proof, and it is arguable exactly how strong this evidence is, but it does count as heuristic evidence of some sort.  The fact that we haven't been able to find an algorithm to break AES does seem to count as some sort of heuristic evidence that no such algorithm exists -- again, not ironclad, not a proof, arguable how strong the evidence is, some might even call it weak evidence, but still evidence of some sort.

Moreover, it is easy to prove that if AES is secure, then the function $F$ defined by

$$F(k) = \text{AES}_K(0)$$

is one-way, where $0$ represents the block of 128 zero bits.  To be precise, if AES is a secure PRP, then $F$ is a secure one-way function -- this latter statement can be mathematically proven in a 100% rigorous way.  Here by "security" I refer to concrete security rather than asymptotic security.  If you're interested in asymptotic security, you'll need something more.

Thus, this is heuristic evidence that one-way functions exist: in particular, we have heuristic evidence that the $F$ defined above is one-way.  You can form your own judgement about how persuasive or how strong this evidence is; that probably comes down to a personal judgement.