This article is from the FAQ, by with numerous contributions by others.
1) AT the request of a U.S. government agency (NASA, NSA, CIA, ...)
If a U.S. govt. agency declares an interest in you and petitions
the USIA to waive the HRR. Could be a very good avenue if you work
for that agency or do security-related work.
2) extreme hardship to U.S. citizen or Green Card holder spouse or
child: This is a good option if the hardship is likely to be
extraordinary. Mere desettlement or economic loss may not be sufficient.
3) political persecution; residents of European countries might
not want to pursue this. Residents of the PRC have a blanket
waiver (Pelosi bill?).
4) NORI/no-objection: the home government issues a statement
of no-objection; seems to be the common and successful approach
for Indian citizens.