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Are there any elements that can be connected to the 'cavity' (i.e. the space) under a stud (e.g. on a regular plate or a regular brick)? The cavity measures about 2.4mm in diameter.
So far I've only tried a minifig hand peg (not the clip side, the other side) but unfortunately it is just a bit smaller, it has a diameter of about 2.3mm.
[Edit] I'm talking about the cylindrical cavity with a diameter of 1.5 LU as indicated in the overview posted here -- What are the dimensions of a LEGO brick?.
If I'm understanding you correctly, you're talking about the molding cavity directly underneath the stud (that actually goes up into the stud), and not the main tube cavities under plates/bricks (that are actually offset from the studs).
If that's the case, no, I'm not aware of any element that "intentionally" fits that space.
First off, I don't think that space is made with a high tolerance. Looking at some of my bricks, the dimensions of that cavity are not nearly as uniform as the other features of the brick.
Secondly, I don't know how many System bricks have components that are not integer multiples of LUs.
You might get lucky and get a minifigure piece that has a detail that is 1.5 LUs, but you'd have to try a lot of them.
Without further specification, unless I read your question wrong, ANY normal brick-type element can be connected to the cavity under a stud - the whole principle that LEGO was based on, basically. Anything that has a stud on it will fit.
If you rather meant "non-stud" elements such as hands, signs, rods, etc., please update the question to make it more specific.
(Edit: after re-reading your question I realized I misinterpreted the stud with the bar. Leaving the answer for posterity.)
There is. The size in the small holes beneath newer plates is the same size as a connector that doesn't have a fixed name. Bricklink calls it pin, others call it mini pin or stub. It is used in minifig headgears and a few other places, like the old dragon flame. It's not a connection people often use, but it can be done:
This picture is taken from a small writeup by Brian D'Agostine which documents other uses of it. You can also use the same pin (stub, mini pin, teensy tiny peg) to create fire strands:
Once you're aware of the connection you can find many fun fits for it.
For base plates you can connect the light sabre pieces underneath the base plate but I don't think it was an intentional feature
