Via Tony Hirst I found out about MapZen's elevation tiles, which encode elevation data in PNG and other formats.
These days they live at https://registry.opendata.aws/terrain-tiles/
I managed to download a subset of them using download-tiles like so:
download-tiles elevation.mbtiles -z 0-4 \
--tiles-url='https://s3.amazonaws.com/elevation-tiles-prod/terrarium/{z}/{x}/{y}.png'
I'm worried I may have got the x and y the wrong way round though, see comments on simonw/datasette-tiles#15