Translation:Το πανεπιστήμιό μου
Is το πανεπιστήμιό μου (with two accents) correct?
It's a learnèd word, so I would expect the iota at the end to be syllabic rather than a glide, but you can never tell with Greek spelling.
are the really two accents for emphasis in this word? I was pleased to have spelt this long word correctly and put one accent over the η. Obviously I was disappointed when I got told there was a typo and I had missed an accent over the final omikron. Google translate does not have an accent over the final omikron. What is correct?
Yes -- if a word is accented on the third syllable from the end and is followed by an unstressed word (in modern Greek, this applies mostly to the possessives μου, σου, του, της, μας, σας, τους), then you need an extra accent to present the single accent from falling on the fourth syllable from the end. (Since such unstressed words are pronounced essentially as part of the previous word.)
The two accents are required in words stressed in the antepenultimate syllable when followed by a possessive. This is reflected in speech as well. As troll mentions above, the iota is syllabic, meaning the word is broken down in syllables like this: πα-νε-πι-στή-μι-ο.