What is the title of a person that studies random stuff? or a bit of everything?
A geologist studies the earth.
A _____ studies random stuff.
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A person who derives ideas, style, or taste from a broad and diverse range of sources.
A person of great or varied learning; a person acquainted with many fields of study; an accomplished scholar. [OED]
From Greek πολυμαθής, “having learnt or knowing much.”
smatterer (plural smatterers)
One who smatters; one who dabbles in or experiments with a little bit of everything, especially knowledge.
It depends on what you mean by "study".
A geologist does research on the earth. Someone who "studies random things" who does actual research on all of them would be the most multidisciplinary scientist in the world; perhaps the noun multidisciplinarist can be used.
But more likely, he doesn't do much actual research, but rather reads textbooks, follows news about new discoveries and generally wants to know all there is to know about what actual scientists have found out, on a wide range of subjects. That's a science fan.
If, on the other hand, you're not talking research at all and more about studying as in doing school subjects, then a geologist is someone who did a lot of geology courses. Someone who spread out his coursework over all sorts of subjects is the opposite of a specialist -- a generalist.
To me, it sounds a bit like a TRENDY, if the random subject matters are whatever is popular at the time, it doesn't imply deep knowledge, though it doesn't really imply study either, just following whatever is 'hot' at the moment... When I was in school, everyone was becoming a marine biologist.. that faded and then it was psychology, etc...