This is a difficult question to answer. I cannot give such an average. But what I can do, is speak from experience, and mention what types of people seem to become obsessed, and what their IQ ranges seem to be.

People who, for whatever reason, ceaselessly seek confirmation of their intelligence. This can result from unfortunate family history or lack of opportunity. These people, rightly or wrongly, typically believe they have gifts which exceed what any test has indicated, and accomplishment has sufficiently shown, or any person or group has validated. I think these people likely exist outside of high intelligence groups, but they definitely exist within them. Usually these people have some justification for suspecting they are smarter than they are given credit for. I would place these people at IQ110 plus, SD15.
People who are at the cusp of being accepted into some desirable group they want to belong in. These are people who attempt, but do not necessarily get in, to societies like Mensa or others. E.g. IQ128, IQ133, IQ142, SD15 etc.. These are all just below certain IQ society entrance requirements.
People actually in the high IQ societies. You might suspect that membership in these groups would be confirmation enough, but in reality, there is a competition and desire for increased prestige in these societies. Not everyone participates in this way, but some do. These are in the group IQ120+SD15, 90th percentile (the requirement of the lowest level IQ society).
People who are outliers and score extremely high on intelligence tests. >IQ145 SD15 by subtest, >IQ160SD15 by FSIQ. These people often wonder what they are capable of. These people often have ceiling scores on tests, and literally do not know what their maximum abilities are. They are often interested in the topic for justifiable reasons, and they make contributions to psychometrics and even create intelligence tests for others. They are not common enough to contribute significantly to the average, but I include them here because they are often quite preoccupied, focused, or interested in these tests.
I would have to say that I did not witness obsession or preoccupation with IQ until I was in the high IQ societies. Membership alone invites continual communication about this topic, and the topic of giftedness in general. My guess, which I am reluctant to give, is that the average of people obsessed with IQ for long periods of time (not just for a month or two) is greater than IQ115, probably approaching 130. At universities and at jobs requiring high intelligence, people simply do not talk about it much. However, perhaps people are just not open about their obsessions. Nevertheless, I suspect people who are truly obsessed discuss it frequently, and I did not see that until I was a member of Mensa.

The average of people who are interested in IQ (which seems to be everyone, except on the lowest end), is probably around 105 - 110.
