Take a K3, or a general $n$-dimensional complex torus $M$, $n>1$, without any integer (1,1)-classes, and remove a point. You will obtain a non-compact Kahler manifold without any non-trivial line bundles (because its second cohomology stays the same). For such a manifold it's not hard to show that no positive bundles exist. Indeed, if there is a positive line bundle, it must be trivial. Then you would have a positive form $\eta$ which is exact. By Skoda-El Mir theorem, the trivial extension of $\eta$ to $M$ is closed, positive and hence exact current. This is impossible, because $M$ is compact and Kahler.

Take a K3, or a general $n$-dimensional complex torus $M$, $n>1$, without any integer (1,1)-classes, and remove a point $x$. You will obtain a non-compact Kahler manifold without any non-trivial line bundles (because its second cohomology stays the same). For such a manifold it's not hard to show that no positive bundles exist. Indeed, if there is a positive line bundle, it must be trivial. Then you would have a positive form $\eta$ which is exact. By Sibony's lemma (see e.g. arXiv:0712.4036, Theorem 5.1), $\eta$ is locally integrable around $x$. Then, by Skoda-El Mir theorem, the trivial extension of $\eta$ to $M$ is a closed, positive and hence exact current. This is impossible, because $M$ is compact and Kahler.