What would you call a person who is not a student, but takes interest in exploring academic topics?

Solution 1:

How about independent scholar?

Solution 2:

Actually, the word "student" need not mean "one who is enrolled in formal studies," but can mean only "one who studies."1 This latter sense can be invoked by saying, "a student of X." When you call someone just "a student," though, you do say that they are enrolled somewhere.

Though wholly without formal education, he was a keen student of history.

Solution 3:

You can use dilettante, but it sounds kinda derogatory.

In the engineering fields, enthusiast has become quite common too; see here for an example.

Solution 4:

Are you looking for autodidact?

The first part auto means 'self', and didact means 'taught', so an autodidact is one who is self taught.

Solution 5:

There are several senses of scholar that fit the bill:

One who educates themself for their whole life.

One who learns anything: as, an apt scholar in the school of deceit.

One engaged in the pursuits of learning.