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](http://www.dol.gov/whd/regs/compliance/whdfs71.pdf) 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.