If you keep the payload under 4 pounds I believe it's theoretically unregulated in the US except section 101.7 - don't create a hazard. I would still try to be approximately in compliance with part 101 though. Not a laywer.
As rules go, IIRC at least here in the Czech Republic anything below 100 g is unregulated, you just need to have a licensed amateur radio operator, so that the probe can broadcast in the 20 meter WSPR band under his callsign.
Plus, if @jacquesm says it, it must be true. ;-)