  In the article, the rough path theory is extended to cover paths from the
exponential Besov-Orlicz space \[B^\alpha_{\Phi_\beta,q}\quad\mbox{ for }\quad
\alpha\in (1/3,1/2],\,\quad \Phi_\beta(x) \sim
\mathrm{e}^{x^\beta}-1\quad\mbox{with}\quad \beta\in (0,\infty),
\quad\mbox{and}\quad q\in (0,\infty],\] 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).
