Nazi scientists made significant progress in rocketry. Wernher von Braun, who is considered a Father of Rocket Science, was the designer of the Nazi's V-2 rocket. Although the Nazis were less interested in using rockets for space flight and more interested in using rockets as long-range missiles (in replacement of bombing raids).

After the war, Von Braun worked for NASA and would later be one of the lead designers of the Saturn V rockets.

Don't know if this counts as an invention, but the torch relay in the Olympics was introduced in the 1936 Olympic games.

Mod note: please make your answer comprehensive and informative. Don't just shout FANTA.

The first practical helicopter, the Focke-Wulf Fw 61, was made in Germany in 1936. Interestingly enough one of the test pilots was Hanna Reitsch, a rather devoted Nazi who flew her plane into Berlin during the during the Soviet attack on the city, and was present in the Fuhrerbunker just days before the end of the war. She and two others actually flew back out of the city from an improvised airstrip during the height of the battle and surrendered to the Americans a few days later.

While most of the "medical experiments" performed the Nazi's performed on prisoners are considered scientifically useless, the data generated by their research on hypothermia is considered by many to be valid.
Of course, the research itself was unequivocally brutal and criminal, so debate continues on whether the data can be trusted, or if using the data at all is ethical.

EDIT: to be clear, I take no position on whether the results of Nazi hypothermia research is or is not valid. There seems to be disagreement on the matter, while, as far as I know, all other nazi 'medical experiments' are considered garbage.

Adolf Butenandt was the first to isolate & identify the sexual hormones oestrogen (1929), progesterone and androsterone (1931) which led to the invention of hormonal birth control.

Can you clarify what you mean by life jackets? Because if you just mean in general, rather than some specific type, then they were around quite a bit earlier, featuring in any description of the Titanic for example.

There were a few things that would have promising results later, but the tech wasn't there yet to fully realize their potential.

• Zielgerät 1229, an active infrared scope--in other words, night vision. It weighed 5 lbs plus a 30 lbs battery. Being deployed in small numbers in early 1944, it was a case of too little, too late.
• Focke-Achgelis Fa 269, a tiltrotor aircraft. This allows the propellers on the wings to tilt vertically so it can take off and land like a helicopter, and then return to horizontal to fly like an airplane. The German prototype never flew, but today the basic design has been realized in the American V-22 Osprey.

This really depends on your definition of 'Nazi' and 'invent', since most things Germany created had parallel developments in other countries and it was a series of inventions that evolved over time, so the 'first' can be debated.

Jet engines were a major development, but Britain also developed them and whether you count the Germans or British as first depends on what milestone you choose.

There are still some laws in force from that time e.g. laws about nationality. Like The principle of inheriting the german nationalityfrom the parents instead of the country is birth.

Source: I'm a german lawyer