Hard to pin down the object I find the most intricate and beautiful mathematical structure, but in my opinion this structure has yet to emerge from the growing body of our knowledge. I believe a structure mathematicians are gradually understanding is manifesting itself in various areas of algebra, number theory, analysis, probability theory, theoretical physics, geometry, topology, set theory and several others, and one might thus say that the most intricate and beautiful structure in mathematics is mathematics itself. More precisely it is some unknown object that all mathematicians study from different sides, angles and perspectives without yet knowing the name for the thing that their discoveries actually have in common. Some evidence to this is given by repeated patterns occurring in several answers here, as well as by some striking instances of partial unification of mathematical concepts, such as (to name very few) Connes' noncommutative geometry, Arakelov theory, Segal's modular functors, topos theory and homotopy type theory.