I wasn't able to find a single example of more than two propellers stacked axially in a contra-rotating configuration. There is a huge number of contra-rotating propeller setups, but it is always a combination of two propellers on the same shaft, not four or six.
Why isn't a piston engine used to run more than two standard propellers on a single shaft? Hasn't it been tried (or tested at least) in 100+ years of aviation history?
(This question isn't about turbofans/turbojets, but I also couldn't find anything for ducted fans: two props on a single axis is a maximum.)
Blade-solidity is explained here:
Contra-rotating props explained here:
An increase in a number of blades does seem to reduce the efficiency, whereas an increase in propellers on an axis seems to increase the efficiency.