Enhancing the fidelity of weather and climate simulations requires capturing smaller scales and complex atmospheric processes. However, the computational demand for such precision easily exceeds the capabilities of current supercomputers, including modern exascale machines. We address this challenge by investigating advanced numerical concepts that significantly improve the effective resolution of atmospheric simulations, namely dynamic Adaptive Mesh Refinement (AMR) and Discontinuous Galerkin (DG) methods. Adaptive methods allow for a continuous readjustment of computational focus in time and space. This increases the achievable level of detail while reducing time-to-solution and resource consumption. However, they require a sophisticated selection of adaptation criteria, memory layouts, and communication patterns to fully exploit modern HPC infrastructures. High-order DG methods offer high accuracy even on coarse meshes, keeping data requirements low and resulting in exceptional efficiency. Furthermore, their high arithmetic intensity is primarily local, lending the algorithms to massive parallelization. While DG methods have historically been susceptible to numerical instabilities, recent advancements, particularly the development of entropy-conserving schemes, have significantly mitigated this issue. By incorporating a physics-based concept of stability, based on the thermodynamic principle that entropy cannot shrink, spurious disturbances are effectively ruled out. Despite their potential, neither AMR nor DG has been widely adopted in global circulation models. We aim to bridge this gap through the development of an exascale-ready simulation framework with the intention to support more complex scientific investigations on climate time-scales. In particular, we combine a highly scalable mesh management library t8code, a dedicated atmospheric flow simulation software TrixiAtmo.jl, written in Julia, an interface library enabling legacy Fortran codes to leverage Julia packages, and MESSy, a software framework for building global and regional chemistry-climate models. We will outline the interplay of the different software frameworks, present results for classical benchmark problems and scaling analyses, and discuss current challenges, such as the efficient utilization of modern GPU-based supercomputers.