No the auto_increment just keeps on counting over time. Indeed, that value can get quite high over time but the maximum value of an int is ~2 billion (~4 billion if you make it unsigned) so that should most likely suffice.
If you're wondering why id's don't get "recycled", consider a scenario with two tables, let's say student and grades, are linked via id's. Let's now say you remove a student but not the grates that belong to the student [from the grades table].
If a new student that enrolls were to get the recycled id from the student that was just deleted it would automatically get the grades of that student, and that's not what you'd want to happen