After processing complete, SSAS DB slowly consumes all available space on disk, w. high memory use also
I found the answer eventually ....

There was a job set up to incrementally process the cube. Incremental cube processing requires a query to be defined to select new fact data to merge into the cube. In my case, this query was defined badly and was selecting the entire set of fact data, i.e. all the data that was already in the cube. Although fully processing the cube did not consume excessive disk space, for some reason incremental processing consumed all the available space. As a result the process never completed successfully. Rebooting the server caused the files created by the unfinished process to be deleted, but then minutes later the processing job would start again.