We define Hilbert function of a semiring ideal of tropical polynomials in n variables. For $n = 1$ we prove that it is the sum of a linear function and a periodic function (for sufficiently large values). The leading coefficient of the linear function equals the tropical entropy of the ideal. For an arbitrary n we discuss a conjecture that the tropical Hilbert function of a radical ideal is a polynomial of degree at most $n − 1$ (for sufficiently large values). For $n = 1$ the conjecture is true, also we have proved it for zero- dimensional ideals and for planar tropical curves.