For various $L^p$-spaces (1 ≤ p < ∞) we investigate the minimum number of complex-valued functions needed to generate an algebra dense in the space. The results depend crucially on the regularity imposed on the generators. For μ a positive regular Borel measure on a compact metric space there always exists a single bounded measurable function that generates an algebra dense in $L^p\left(\mu \right)$. For M a Riemannian manifold-with-boundary of finite volume there always exists a single continuous function that generates...