I just read this inspiring paper: TbGAL about implementation of high dimensions. The key idea is to factorises k-blades as the (outer / geometric) product of vectors, leading to a very performant memory storage. They conducted some test up to dimension 1536. This approach presents 2 drawbacks: handles only versors and k-blades but not general multivectors nor k-vectors that are not k-blades, and the algebra signature should be (p,q,0), but still, the approach is very interesting.
If you have in mind some applications that requires such very high dimensions, just discuss them here.