I do not see any body mentioned Moonshine phenomena. The Fourier coefficient of a modular function gives information about the degree of irreducible representations of Monster group, if I write right. Another object that is deep in my opinion is zeta function. If we write the prime number theorem multiplicatively, which says that the geometric mean of the primes areis $e$, which already exhibitexhibits deep harmony, while the Riemann hypothesis gives much stronger result of the distribution of the primes. Another mystery is that from different point of view, $p$ adically, complex analysis methods or arithmetically, people see the light that zeta functions shed in a unified way.
I would also say reciprocity laws deep, since thousands of different proofs and to really understand often requires new theories like class field theory.
For some others I would say duality theorems in geometry, catalan numbers in combinatorics I will not say anything about this, since I am not a geometer nor a combinatorist and not really understand them well.