Two new classes of compact weighted essentially nonoscillatory (WENO) polynomial limiters are presented for second-, third-, fourth-, and fifth-order discontinuous Galerkin (DG) schemes on irregular simplex elements. The presented WENO-DG procedures are extensions of the high-order WENO finite-volume and finite-difference schemes of Zhu and Shu (2017) [25], (2019) [26] to high-order unstructured DG schemes. A compact positivity preserving limiter is applied to the solutions to ensure pressure and density remain within physical ranges at all time. It is then verified that the bounded WENO-DG maintains the formal order of accuracy of the underlying DG schemes in the smooth regions. The performance of the proposed WENO-DG is also demonstrated with inviscid test cases including the classical Riemann problems, shock-turbulence interaction, scramjet, blunt body flows, and the double Mach Reflection problems.