In the article, Besov-Orlicz regularity of sample paths of stochastic processes that are represented by multiple integrals of order $n\in\mathbb{N}$ is treated. We give sufficient conditions for the considered processes to have paths in the exponential Besov-Orlicz space $$B_{\varPhi_{2/n},\infty}^α(0,T)\qquad \mbox{with}\qquad \varPhi_{2/n}(x)=\mathrm{e}^{x^{2/n}}-1.$$ These results provide an extension of what is known for scalar Gaussian stochastic processes to stochastic processes in an arbitrary finite Wiener chaos. As an application, the Besov-Orlicz path regularity of fractionally filtered Hermite processes is studied. But while the main focus is on the non-Gaussian case, some new path properties are obtained even for fractional Brownian motions.