Even though the persistence length L_{P} of double-stranded DNA plays a pivotal role in cell biology and nanotechnologies, its dependence on ionic strength I lacks a consensual description. Using a high-throughput single-molecule technique and statistical physics modeling, we measure L_{P} in the presence of monovalent (Li^{+}, Na^{+}, K^{+}) and divalent (Mg^{2+}, Ca^{2+}) metallic and alkyl ammonium ions, over a large range 0.5 mM≤I≤5 M. We show that linear Debye-Hückel-type theories do not describe even part of these data. By contrast, the Netz-Orland and Trizac-Shen formulas, two approximate theories including nonlinear electrostatic effects and the finite DNA radius, fit our data with divalent and monovalent ions, respectively, over the whole I range. Furthermore, the metallic ion type does not influence L_{P}(I), in contrast to alkyl ammonium monovalent ions at high I.