Does there exist something that is possible but not conceivable?
It's not obvious at all to me that possibility is different than conceivability, and your comments on the matter are presently insufficient to show this.
1. Your definition of 'possibility' begs the question to say it is independent of persons. I see no reason to believe why the 'possibility' should be concerned with anything more than simply ways in which the world might exist. It also assumes a very objective sense of the term 'world'. But all I can assume is that the world is simply all of my perceptions. I have no evidence of anything independent of my perceptions. So it is possible to say that the ways in which the world might exist are simply which ways my perceptions are limited.
"The latter is limited by an individual's mental abilities."-Sure, but what if my mental abilities are equivalent to how the world can behave (akin to what I just said a moment ago)
So, it's certainly not obvious to me that they are both different.
2.
"Obviously one can't list something that one believes is possible but not conceivable, because by imagining it, one can conceive of it. It's simply asking someone to do something they are not capable of doing."
This misses the overall point. At best, if your conclusion here follows, all it shows is that one cannot generate a concrete counterexample. However, this does nothing to preclude the possibility of a proof of mere existence, that has no concrete referent. It also does nothing to preclude a proof by contradiction. However, at least you note later that " it could be provable that something isn't conceivable on purely logical grounds somehow,"
However, your justification for this is not nearly so inscrutable as you suggest. This is mainly because a mere claim of the person who disputes such a logical argument is insufficient to rebut that argument (because such a claim would require actual evidence in order to shift back the burden of proof; a claim alone won't rebut that argument). So we don't have to worry about potentially not knowing whether or not that particular person truly could "conceive of it". The keyword here is 'it', which betrays your error of supposing a logical argument would have to provide some concrete possibility, the 'it'.