One of mine is a Fae character. Now, generally speaking, most campaigns are lower level, so the ECL caused by being even part Fae usually puts the notion of a Fae character out of the question. Depending on design as well, a GM might be (rightfully) wary about certain Fae characteristics, namely their spell-like abilities and damage resistance, where applicable. Flight can also be a factor, for Fae that fly. Flying can allow for creative bypasses of barriers otherwise present, which again make a GM's job harder in finding the challenge to provide to a Fae character. Then there may also be a size factor.
But, with that aside, what do I mean by Fae, and what do I intend by the notion of playing one? The answer is that I find Fae lore intriguing, and I know this varies by setting as the GM designs. Consider a race like my display picture, a Sylph. By technicality, not exactly a Fae, but it wouldn't be hard to identify a similar being that is one. As a Fae, one possible underlying motivation behind adventuring with non-Fae is exactly because they are non-Fae, and generally not incorporated into Fae Courts. Maybe it's one that has never seen a human in person before. It's a strange form of curiosity, and, let's be honest here, I tend to like strangeness in my characters. Maybe they're 'on vacation', as they put it, from the Courts in a setting where there are immense political conflict between the two. Maybe they're an emissary of the Fae in the non-Fae world. Or maybe it's actually a Fae adventuring group. A lot of possibilities that could be played with.
I love the idea of flying, and I like small characters. A Faerie character is one possibility where I could play with both aspects.
The other is so far removed from the common fantasy setting that a setting in which it is not only applicable, but would be able to fill a substantial role in an adventuring party, is rare. This is a class by the name of Cybernaut, from a supplement to the D20 Modern system.
To put it simply, in the D20 Cyberscape setting (or a setting that would apply to Cybernaut), the internet is upgraded to a virtual reality that 'overlays' the real one. Cybernauts are the experts at interfacing with that virtual reality, and have powerful avatars (programs that act as their intermediary with that virtual world). Naturally, their skillset includes programming and hacking.
This sort of recursion, where a character has basically another character, can be rather confusing, and is undoubtedly part of the reason why such a setting would not be common. It is also rather niche, with one of the main issues being that an entire party of Cybernauts would undoubtedly have too much overlap to be satisfying to the players, but a lone Cybernaut in a party more focused in the 'real world' can also be unsatisfying, not having many situations to shine in. The dual worlds is another reason why it would not be common - GMs usually have enough on their plate handling one.
By consideration of that last sentence, roughly speaking the Cybernaut is an archetype that could potentially apply to any dual-world scenario. However, such scenarios are very uncommon, and modern/futuristic campaigns are not very commong in the first place. Basically, the setting would need to be made to accomodate them in the first place, and that is something that puts more work onto the GM than might be reasonable if they didn't intend for that sort of setting. I know I've had trouble trying to figure out a hypothetical setting in which it'd work without major modifications to mechanics in the printed book (which is another thing that GMs in general are wary of, for good reason).
So there you have it, those are my two primary concepts that I like but probably won't be able to play (or at least not often). There are others as well; I've made multiple tentative attempts at custom classes before, as Sib can attest, but those two are the main ones I return to thinking about more often.