Turn the types up slowly

This project has types switched on loosely. I want strict, eventually.

- Turn on one compiler flag at a time. Tell me the order, easiest first.
- For the flag we're doing now, show me the errors it will produce and
  fix them in batches by kind, not file by file.
- Where a real fix needs restructuring, say so and leave a marked
  escape hatch rather than pretending with `any`.

Do not add `any` to make an error go away without telling me.