Does every finitely generated group have a maximal normal subgroup?

If you mean nontrivial maximal normal subgroup (not 1 or the whole group), then the answer is no.

Higman constructed a finitely generated infinite group $G$ with no subgroups of finite index. You then get a finitely generated group with no nontrivial normal subgroups by taking the quotient by a maximal normal subgroup.

Higman's group $G$ is $\langle a,b,c,d | a^{-1} b a = b^2, b^{-1}cb = c^2, c^{-1}dc=d^2, d^{-1}ad=a^2 \rangle$

See Higman, Graham. A finitely generated infinite simple group. J. London Math. Soc. 26, (1951). 61--64.

Edit:

If you mean does it have a proper maximal normal subgroup, then the answer is yes:

Finitely generated groups have a (possibly trivial) maximal normal subgroup. Higman's reference for this is B.H. Neumann, "Some remarks on infinite groups ", Journal London Math. Soc, 12 (1937), 120-127.

Check out the Tarski monster. It is 2-generated and simple.
Unless I misunderstood your question and you exclude infinite simple groups altogether.

So many answers! I'm completely lost. The paper of "B.H. Neumann, "Some remarks on infinite groups", Journal London Math. Soc, 12 (1937), 120-127" stated results for the existence of maximal subgroups, not maximal normal subgroup. Is this existence question of nontrivial normal subgroup still unsolved?