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Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although the Betti numbers of the Milnor fiber F can be expressed in terms of the jump loci for rank 1 local systems on the complement, explicit formulas are still lacking in full generality, even for b1(F). We study here the "generic" case (in which b1(F) is as small as possible), and look deeper into the algebraic topology of such Milnor fibrations with trivial algebraic monodromy. Our main focus is on the cohomology jump loci and the lower central series quotients of π1(F). In the process, we produce a pair of arrangements for which the respective Milnor fibers have the same Betti numbers, yet non-isomorphic fundamental groups: the difference is picked by the higher-depth characteristic varieties and by the Schur multipliers of the second nilpotent quotients. [ABSTRACT FROM AUTHOR]