I came across this question while preparing for my logic exam.
Can this class be (finitely) axiomatizable, where the class contains all structures $\mathfrak{A} = (A, <, f)$, and for no $a \in A$ is $a < f(a) < f^2 a < ... $.
I believe that this class is not axiomatizable owing to Löwenheim-Skolem (if there exists an random infinite model for a set of sentences, there also exists a finite model.)
I'm not sure how you intend to get to a disproof from your observation.
This has an easy solution via the compactness and completeness theorems, though. Let $T$ be the any first-order theory for which every structure in the class models $T$.
For every chain of inequalities of finite length
$$ a < fa < f^2a < \ldots < f^n a $$
we can find a finite ordered set that models this chain and is in the class, and thus this statement must be consistent in $T$.
Because every finite subset of the infinite chain is consistent with $T$, the entire infinite chain must also be consistent with $T$, and therefore have a model. Therefore, $T$ must also have a model that is not in the class.