Is every holomorphic line bundle on the  say  punctured unit disc $\dot{\Delta} \subseteq \mathbf{C}$ trivial? GriffithsHarris (p. 39) prove that $H^{p,q}_{\overline{\partial}}(\Delta) = 0$ (for $q \geq 1$), and mention that by replacing discs by annuli that one could prove also $H^{p,q}_{\overline{\partial}}(\dot{\Delta}^k \times \Delta^\ell) = 0$. They seem to imply that in particular $H^{p,q}_{\overline{\partial}}(\dot{\Delta}) = 0$. If this were true, using Dolbeault's theorem and the Kummer sequence one could conclude $H^1(\dot{\Delta},\mathcal{O}^\times_{\dot{\Delta}}) = 0$, hence that every holomorphic line bundle on $\dot{\Delta}$ is trivial.
Yes, every holomorphic vector bundle of any rank is trivial on the punctured disk $\dot{\Delta}$ . Indeed, since $\dot{\Delta}$ is a Stein manifold ( like any noncompact Riemann surface ! ) the Oka metaprinciple (here a Theorem of Grauert ) says that the classification of holomorphic vector bundles on that manifold is the same as that of topological vector bundles. But since the punctured disk is homotopically equivalent to a circle , all topological complex vector bundles are trivial, hence the triviality of all holomorphic vector bundles. (Do not confuse with the Möbius vector bundle, which is a nontrivial real vector bundle!) Bibliography If you want to read complete proofs, you can consult Otto Forster's Lectures on Riemann Surfaces. 

