Probably find new questions. Maybe even make them up entirely.
Humans are pretty contrary - and really don't like being told something can't be done. So, they'll reach the end of any set of questions, and find new questions to ask. They'll look at their mapped out rules and laws for what makes things possible, and figure out how to break them - deliberately setting themselves out to find ways to do impossible things. They will continue asking impossible, un-provable questions because why not, and believe things because they believe them, not because they know or can prove anything.
Other possibilities - people forget. It's quite possible that after your theoretical everything is known, people will look back at the historical record and find things that have been overlooked, forgotten, or not recorded for some reason, and set out to rediscover them (reenacting, kinda). If the problem of no questions is pervasive enough, there might be social experiments deliberately losing information and recreating former historical societies to keep people searching. The ultimate cyclical question - how did we ever get along before we knew "X"?... lets find out!
And the last possibility... people are kinda crazy. I doubt, beyond all doubts, there will ever come a time where we understand people, and have run out of questions about us, because people don't run on logic (look at fanatics and extremists who don't even follow self-preservation). And even if we spend all of eternity trying to figure out the biological roots of human-people behavior (and I really doubt it will work, but whatever), we will still have plenty of questions about how other people will react - alien species, or species we uplift or create just because (how else will we answer those questions?), and how any of them interact with each other, with us, over time, in the future...
Maybe a species that runs on logic would be able to ask and answer all of those logical questions, and end up in a crisis because they don't know what else to do, to be, to discover. But we are creatures of instinct and impulse, no matter how much we like logic, and we are not gonna stop until the crazy runs out.