Motivated by recent experimental realizations of topological edge states in Su-Schrieffer-Heeger (SSH) chains, we theoretically study a ladder system whose legs are comprised of two such chains. We show that the ladder hosts a rich phase diagram and related edge-mode structure dictated by choice of interchain and intrachain couplings. Namely, we exhibit three distinct physical regimes: a topological hosting localized zero-energy edge modes, a topologically trivial phase having no edge-mode structure, and a regime reminiscent of a weak topological insulator having unprotected edge modes resembling a twin-SSH construction. In the topological phase, the SSH ladder system acts as an analog of the Kitaev chain, which is known to support localized Majorana fermion end modes, with the difference that bound states of the SSH ladder having the same spatial wave-function profiles correspond to Dirac fermion modes. Further, inhomogeneity in the couplings can have a drastic effect on the topological phase diagram of the ladder system. In particular for quasiperiodic variations of the interchain coupling, the phase diagram reproduces Hofstadter s butterfly pattern. We thus identify the SSH ladder system as a potential candidate for experimental observation of such fractal structure.