Let $X$ be a complex manifold with complex dimension $d$ and structure sheaf $\mathcal{O}\_X$. Let $E$ be a locally free sheaf on $X$. A $holomorphic$ connection on $E$ is a morphism of sheaves
 $$\nabla: E \to E \text{ }\otimes\_{\mathcal{O}\_X} \Omega\_{X}^{1} $$ satisfying the product rule $\nabla(fs) = s \otimes df + f\nabla(s)$ for all open $U \subset X$ with $f \in \mathcal{O}\_X(U), s \in E(U)$ . The connection $\nabla$ is said to be $flat$ or $integrable$ if the composite $$ E \xrightarrow{\nabla} E \text{ } \otimes\_{\mathcal{O}\_X} \Omega\_{X}^{1} \xrightarrow{\nabla\_1} E \text{ } \otimes\_{\mathcal{O}\_X} \Omega\_{X}^{2}$$ is $0$ where $\nabla\_1$ is the map to 2-forms $s \otimes w \mapsto \nabla(s) \wedge w \text{ } + s \otimes dw$. What is a simple example of a triple $(X, E, \nabla)$ with $\nabla$ non-integrable? Any such example must necessarily have $d \geq 2$. This is crossposted from SE [here](https://math.stackexchange.com/questions/3671210/simple-example-of-non-integrable-holomorphic-connection). From that discussion it is apparent that integrability is a strict condition, as even the existence of a holomorphic connection on a holomorphic vector bundle implies vanishing of Chern classes in the compact Kahler case, and integrability is a still stronger condition, but I have not found an explicit example.



Simple example of non-integrable holomorphic connection
Probably this answer intersects with the previous ones. Consider the complex Heisenberg group $H$ of the $3\times 3$ complex matrices with 1 on the diagonal and 0 under the diagonal. Let $x, y, z$ be the other entries, $z$ being in the corner. The center $Z$ is $\{x=y=0\}$. One has the (trivial) line bundle
$$Z\to H\to H/Z\cong C^2$$
The left-invariant $2$-plane field $\nabla$ whose value at $I$ is $z=0$ is a non-integrable holomorphic connection on this bundle. If one insists that the manifolds be compact, one just has to quotient $H$ on the right by the lattice $\Gamma$ of matrices with entries in $Z[i]$. Let $E:=C/Z[i]$.
One has the bundle
$$E\to (H/\Gamma)\to E^2$$
and the image of $\nabla$ is a non-integrable holomorphic connection on this bundle.


Another famous explicit example is the hyperplane distribution $\nabla$ orthogonal
to the complex lines in $C^{n+1}\setminus 0$, which is a holomorphic connection for the tautological line bundle over $CP^{n}$, and not integrable for $n\ge 2$.


