Joseph Priestley discovered the chemical compound in 1772 and in 1799 Humphry Davy discovered, that it can be used medically.
No one "invented" laughing gas because it is a chemical compound known as Nitrous oxide. The first person to synthesize nitrous oxide was the English chemist Joseph Priestley in the mid-1770's. Humphry Davy, another chemist and inventor, discovered its anesthetical applications in the 1790s.