Motivated by the classical Theorems of Picard and Siegel and their generalizations, we define the notion of an essentially large effective divisor and derive some of its arithmetic and function-theoretic consequences. We then investigate necessary and sufficient criteria for divisors to be essentially large. In essence, we prove that on a nonsingular irreducible projective variety $X$ with $\mathrm{Pic}(X) = \mathbb{Z}$, every effective divisor with $\operatorname{dim}X + 2$ or more components in general position is essentially large.