According to you, what are the deep reasons, at a very fundamental level, that makes the notion of compactness so useful and ubiquitous throughout modern mathematics: theory of compact groups, compact Riemann surfaces, classification of compact topological surfaces, Stone-Weierstrass and Ascoli in analysis... and I'm sure I'm forgetting a dozen of applications of compactness in diverse areas of mathematics..

Somebody told me recently that for him, the basic reason for compactness to be so powerful is that in a sense (which he did not make explicit) a compact space "contains all its infinities". It's a bit like what happens with projective spaces; those spaces are built to "contain all their infinities", which make them more powerful than affine spaces. Would you agree with that ?