LLVM has a back end for both AMD and NVIDIA GPUS. Is it currently possible to compile c++ (or a subset) to GPU code with clang and run it? Obviously things like the standard library would be unavailable, as well as operator new and delete. I'm not looking for OpenCL or CUDA, I'm thinking of a fully ahead-of-time compiled program, even a trivial one.
No, you need some language like OpenCL or CUDA, because a GPGPU is not an ordinary computer and has a different programming model (grossly speaking, SIMD like). GPGPU compute kernels have specific constraints.
You might want to consider using OpenACC pragmas in your C++ code (and use a recent GCC compiler).