Say you have 2 operators (can be assumed observable), does $[A,B]$ imply that either $B = f(A)$ or $A = f(B)$ for some function $f$?

This is true in the case of a spin-1/2 Hilbert space, as any observable can be decomposed in a linear combination of the 3 Pauli matrices and the identity, and therefore if $A = \sigma_z$ then $B$ cannot contain neither $\sigma_x$ nor $\sigma_y$.

Another trivial case is if $B = 1$ in any Hilbert space, since it can be written as $B = A^0$.

I assume the first step to proving this (assumed true) would be to consider their shared eigenbasis.