Let $A$ be a finite dimensional commutative algebra. We can assume that it is local. 



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> Question: Which such $A$ have the property that every finite dimensional $A$-module has complexity at most 1? (This should be equivalent to the simple module having complexity equal to one or equivalently bounded Betti numbers)
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Note that a module has complexity at most one if and only if the terms in a minimal projective resolution have bounded dimensions.


Examples include $A=K[x]/(x^n)$. Do you know other examples?



Commutative algebras with modules of small complexity
There are no other examples. This property is equivalent to $A$ being a hypersurface (see Avramov's note "Infinite Free Resolutions"). By Cohen Structure Theorem, an Artinian local hypersurface (which is automatically complete) that contains a field must be isomorphic to $k[[x]]/(f)$, which is equal to $k[[x]]/(x^n) = k[x]/(x^n)$ with $n$ being the smallest power of $x$ in $f$. 


