How many years old is the oldest continuously operating university?

How to get close to this number

The word "continuously" is doing the work. Older institutions of learning existed and closed; the question asks for one that has never stopped. That requirement excludes almost everything ancient and points at the medieval Islamic world, where endowed institutions had legal structures that survived dynastic change.

The comparison in the explanation is the useful one: Europe's oldest dates from two centuries later. The mental default places the origin of the university in medieval Europe, and the actual answer is North Africa — which is the same geographic correction as the cave painting question on this site.

The founder detail is worth stating plainly because it is so rarely known: it was founded by a woman. The oldest continuously operating degree-granting institution in the world was established by Fatima al-Fihri in the ninth century, which sits awkwardly against almost every assumption people carry about the history of formal education.