In the article, the rough path theory is extended to cover paths from the exponential Besov-Orlicz space \[B^α_{Φ_β,q}\quad\mbox{ for }\quad α\in (1/3,1/2],\,\quad Φ_β(x) \sim \mathrm{e}^{x^β}-1\quad\mbox{with}\quad β\in (0,\infty), \quad\mbox{and}\quad q\in (0,\infty],\] and the extension is used to treat nonlinear differential equations driven by such paths. The exponential Besov-Orlicz-type spaces, rough paths, and controlled rough paths are defined and analyzed, a sewing lemma for such paths is given, and the existence and uniqueness of the solution to differential equations driven by these paths is proved. The results cover equations driven by paths of continuous local martingales with Lipschitz continuous quadratic variation (e.g.\ the Wiener process) or by paths of fractionally filtered Hermite processes in the $n$\textsuperscript{th} Wiener chaos with Hurst parameter $H\in (1/3,1/2]$ (e.g.\ the fractional Brownian motion).