In episodes #2.10 through #2.13 Jeannie wanted to know her birthday. The Show had a contest to guess the month and day. It started with "The Girl Who Never Had a Birthday, Part 1" (#2.10), wherein her year of birth was determined to be 64 BC. As these episodes take place in 1966, Jeannie is 2,029 years old. The actual month and day were finally revealed in "My Master the Great Caruso" (#2.13): Jeannie was born on April 1. April Fools Day, naturally.
Earlier in the series, her birthday was said to be July 1st.