We present AMHOEBA, a software suite for algorithms that combine higher-order cut cell discretizations with adaptive mesh refinement for a variety of application domains. To help deal with the complexity of embedded boundary (EB) data, we use distributed hashes to manage sparse indexing into spatial locations, refinement levels, and complex geometric information. Features include finite volume treatment of multiple materials and physics, as well as block-adaptive mesh refinement and mappings. Special support is available for merging regular grid discretizations with sparse matrix operations on subsets of the domain; we call these "hyper-sparse" operations since they only apply boundary conditions and irregular stencil operations in narrow regions, thus greatly reducing the memory needed, compared to sparse matrices across the full mesh. The software is distributed with SHMEM and supports accelerators, so that after initial setup most of the computation and communication is directly on the GPU. We demonstrate the approach with several test problems for model equations with higher-order space-time discretizations. We also show how very accurate solutions can be obtained for Maxwell's equations with complex material interfaces, and for moving boundary solidification problems. Overall, we show that AMHOEBA makes it possible to compose complex higher-order EB algorithms into fully accelerated, highly accurate application simulations.