As my MSDN Azure subscription was threatening to have exhausted its credit before the end of the month, I decided to see if it was easy to resize the VM, or at least the Blobs making up the disk storage. Those disks are SSDs, 1TB. Azure charges for this by disk size (so called Premium Data Storage), and not by the actual data used. It is not possible to EASILY resize those disks, so I decided cut my losses and create a 128GB SSD as my data disk, which should easily be enough for my needs.
As so often, everything else in the post is a set of screenshots: