(Joint with Samuel Grushevsky, Gabriele Mondello, Riccardo Salvati Manni) We determine the maximal dimension of a compact subvariety of the moduli space of principally polarized abelian varieties A_g for any value of g. For g<16 the dimension is g-1, while for g>=16, it is determined by the larged dimensional compact shimura subvariety, which we determine. Our methods use functional transcendence theory.