What are some examples of algebraic varieties over the complex numbers with no (algebraic) vector bundles other than the trivial ones?
The only example I can think of is $ \mathbb{A}^n_{\mathbb{C}} $ and this is a very deep example - see the Quillen-Suslin theorem.
The inspiration for this question is the following fact that I learnt quite late in life: The 3-sphere $ S^3 $ does not have any topological real vector bundles other than the trivial ones. In fact, the excellent answer to this question tells us much more.