2 of 2 added 453 characters in body

That would be illegal (for the supervisor/university) in the US.

Edited by popular request to say that like an unpaid internship the US Department of Labor has some pretty strict rules for when an "employer" is allowed to not pay someone who works for them. At my university, postdocing is work, so I think that role should comply with the same rules. At my university a postdoc wouldn't fit either, so in my experience, unpaid postdocs are extremely uncommon.