I was under the impression that it was Bologna, followed by Oxford, followed by Cambridge. But I've also heard recently that Morocco and Mecca both have much older universities, from the 9th century. Is that true? Is this a matter of "what exactly is a university?"
It's a question of "what exactly is a university." The Islamic world in about the 10th century saw a huge proliferation of "Madrassas", more specifically Nizamiyya Madrassas, after one of the patrons of their spread, the Seljuk Vizier Nizam al-Mulk. The idea grew so popular that, by the 11th century these Madrassas had transcended even the Seljuks own borders so that, for instance, the Fatimids and Ayyubids in Egypt copied the same model.
The "university" of Qaraouine in Morocco is a similar kind of school pre-dating this trend, as indeed there were (now destroyed) such examples in Persia that served as a model to Nizam al-Mulk in the first place.
I bring up this spread of Madrassas because some scholars have posited that it seems awfully coincidental that you have an 11th century spread of institutions of tertiary education in the Islamic world followed by the spread of Universities in Europe in the 12th.
While I'm sympathetic to this view it is essentially one based almost entirely on coincidence (one of the founders of the University of Paris having lived in the Levant notwithstanding).
While it's important to recognize the spread of tertiary education in the Islamic world, these madrassas differed in important ways from universities as they have come to be understood in the western world. While a religious education was at the front and center of both a Madrassa and a University, only at universities do you develop "secular" fields of study like civil law and mathematics. Madrassas would have trained their students to a wider range of studies than a mere "theology" curriculum at a western university but that's because religious studies in medieval Islam encompassed various fields that in Europe would have been seen as mostly secular pursuits (i.e. grammar, or the law).
Perhaps someone with a greater expertise in European universities can comment on how and when these "secular" fields of study became fully fledged parts of a university education (and I put these and other terms in quotes again because we're talking about, even for math students, a course of study that would have been incredibly intertwined with religious study and practice, nor would the people taking these courses have conceived of themselves as "secular"). But the overarching point is that these fields of study simply never entered into the curriculum at Madrassas.
So in that sense, we can recognize Madrassas as the oldest continually operating tertiary education institutions in the world, but I would hesitate to call them the oldest "universities."
Rather, the first western-style universities in the Middle East were founded in the late 19th and early 20th centuries when the need for a secular program of study within the countries themselves to educate the populace was recognized prompting the founding of, for example, the University of Cairo in 1908.