gr.group theory - 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 < a,b,c,d | a^-1 b a = b^2, b^-1cb = c^2, c^-1dc=d^2, d^-1ad=a^2 >

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.