The three-dimensional structure of DNA is highly susceptible to changes by mechanical and biochemical cues in vivo and in vitro. In particular, large increases in base pair spacing compared to regular B-DNA are effected by mechanical (over)stretching and by intercalation of compounds that are widely used in biophysical/chemical assays and drug treatments. We present single-molecule experiments and a three-state statistical mechanical model that provide a quantitative understanding of the interplay between B-DNA, overstretched DNA and intercalated DNA. The predictions of this model include a hitherto unconfirmed hyperstretched state, twice the length of B-DNA. Our force-fluorescence experiments confirm this hyperstretched state and reveal its sequence dependence. These results pin down the physical principles that govern DNA mechanics under the influence of tension and biochemical reactions. A predictive understanding of the possibilities and limitations of DNA extension can guide refined exploitation of DNA in, e.g., programmable soft materials and DNA origami applications.