Got rare parts? Post 'em here!
I'll go first! Here's some rare and semi-rare parts I've got:

The two top brackety pieces are the rarest in the picture...the only two in known existence (according to BrickLink). I really got an incredible deal on them at only ~$4 each.!
Some parts are more rare than others, but there are more of them for sale simply because they're not a popular part to build with, while other parts, which may not be as rare, are harder to find (sellers) because they're more popular to build with.
Those two little gun looking pieces on the middle bottom are like that. They were actually in a real Lego set, but you hardly ever see anyone selling them because they're such a popular piece. I think I paid about two dollars apiece for them.
Pieces that are uber tiny are also generally worth more, simply because they're so small they're very likely to get lost in carpets and then accidentally vacuumed up...which means less of them in circulation and consequently higher value.
Some of these are HTF (hard to find) and "test strikes", meaning that they were never actually in any Lego set (in this color). Lego will sometimes make a few thousand of a particular part in a non-set-standard color and will then give them away at conventions, their theme parks, or sell them in their Lego stores; often times in their grab bags. So it's like buying a pack of baseball cards and finding some incredibly rare card.
Such as these lil buggers:

Normally you can only find them in gray (in sets), these were like $2 each I believe.
Different colored cups are probably the most prolific HTF part I've come across:

Most of those have never been in any actual set. Lego produces quite a lot of them though so they're usually not too pricey.
This one is kind of weird:

You wouldn't really think of it as rare, but Lego has never actually made that part in that particular color in any set. I forget exactly how much I spent on it, but I think it was around like $5.
A good example of "rare because they're tiny" parts:

The black ones being the priciest, I think I spent around $10 for the tiny black one.
Here's a good "test strike" example, the red popsicle:

"Test strikes" are most often in red, or milky white, because they're the cheapest colors. Basically they're created when they're testing the machines to see if they're working/calibrated right. The machines run so fast that hundreds are produced in mere seconds. Those that get created usually go to employees who can then sell them online (kind of a nice employment perk).
Lego also makes quite a number of contract sets, for specific outfits. The most notable is the Maersk shipping company, who has contracted Lego to make dozens of different sets for them over the past 30+ years. Those sets contain a special color unique to them called "Maersk Blue", which are obviously rare and obviously quite pricey. Here's a good example:

The Maersk Blue one is in the middle and it's vaguely similar to the newish "medium blue" color.
The most expensive single parts I've bought:

Both the blue and lavender ones I paid about $20 for!
At some point though I want to buy a blue bicycle, which will likely cost me about $60 to $70, so I'll break that record when I do.
I'd also like to get some orange hard hats at some point, which will also cost quite a bit.
Parts that are easily breakable are also generally worth more, like bicycles for example:

The yellow, white, green and pink being the rarest. The pink were in a special contracted set, for some European telecommunications company I think...or something like that.
It should also be noted that just recently some "Medium Azure" bicycles have been made available on BrickLink, from three different sellers. The prices aren't that bad at all either, the cheapest being just $5 a piece!
Last one for now:

The white visors are worth around a buck each and are "test strikes", the brown ones around three dollars each! They were only in like one single Star Wars set. And the blue phone decorated slope is worth a whopping ten bucks just on its own!