= Solution
In the ordinary local two-derivative matter class, the general ungauged <supersymmetric action> for a <chiral superfield> is
$$
\boxed{S=\int d^4x\,d^4\theta\,K(Z,\bar Z)
+\left[\int d^4x\,d^2\theta\,W(Z)+\mathrm{h.c.}\right].}
$$
The <Kähler potential> $K$ is real, and the <superpotential> $W$ is a <holomorphic function>. Adding $f(Z)+\bar f(\bar Z)$ to $K$ changes the <action> only by a <boundary term>: full <superspace integration> of a chiral or antichiral function vanishes modulo that term. This is a <Kähler transformation>. A positive <Kähler metric> $G=K_{z\bar z}$ is required for a nonsingular positive-norm kinetic term, not just for derivative counting.
To see the differential order, the bosonic component Lagrangian in mostly-plus signature contains
$$
\mathcal L_{\mathrm{bos}}=-G\partial_mz\partial^m\bar z
+G F\bar F+W_zF+\overline{W_z}\bar F.
$$
The <auxiliary field> equation is algebraic: $F=-G^{-1}\overline{W_z}$ on a purely bosonic configuration. It gives the <scalar potential> $U=G^{-1}|W_z|^2$. The scalar equations have at most two spacetime derivatives, the <fermion> equations from the sigma-model kinetic term have one, and the remaining interaction terms do not raise this order. The fermionic corrections to the auxiliary equation are still algebraic.
For four-dimensional <renormalizability>, assign <engineering dimensions> $[Z]=1$, $[d^4\theta]=2$, and $[d^2\theta]=1$. Hence, up to a <Kähler transformation> and a constant field normalization, the restrictions are
$$
\boxed{K=\bar Z Z,\qquad
W=w_0+aZ+\frac m2Z^2+\frac g{3!}Z^3.}
$$
A mixed higher-degree term in $K$ or a term of degree above three in $W$ introduces a negative-mass-dimension coupling. The constant $w_0$ has no local effect in global <supersymmetry>, as in the <constant superpotential in global supersymmetry>.
The boxed general-action statement uses the conventional two-derivative matter interpretation. It is not a classification of all derivative-dependent effective <superfield> <actions> by the order of their bosonic equations alone. For example, a scalar Lagrangian quartic in first derivatives still yields a second-order Euler–Lagrange equation, and <supersymmetric> derivative-dependent interactions can contain such terms. Their auxiliary and fermionic equations require separate checking. This distinction is explicit in https://arxiv.org/abs/1012.3748[the original construction of supersymmetric derivative-dependent scalar actions]. Such <actions> are outside the ordinary $K+W$ class used for the renormalizable answer here.
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