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The current airplane wing has to be one of the longer parts at 56 studs.
Largest by volume would be difficult to determine easily. My guess is one of the raised baseplates, probably the one listed below.
According to BrickLink's catalog, the heaviest part with a weight listed is a 100x250 Modulex baseplate at 7300g. The heaviest non-Modulex part is an old 12V train transformer at 1569g. The heaviest non-Modulex part without an electric component is this 24x36 Duplo brick at 880g, though it looks like it wasn't available in a set or individually. Ignoring a 2x8 brick that surely doesn't weigh 475g, it looks like one of the heaviest parts with System studs is this Belville raised baseplate at 330g. In general, the heaviest stuff is train motors/transformers, boat hulls, and Duplo parts.
There are a lot of pieces smaller than a 1x1 round. This is probably one of the smallest individual pieces. But perhaps you're referring to pieces with a stud connection.
these huge boxes are prefabricated elements, that cannot be taken apart.
Well at least they were the largest Lego pieces known to me, if you consider these elements 'pieces', all depending on how you define what a Lego piece is.
So what about raised baseplates?
The problem with the raised baseplates is they are hollow and therefore not much heavier than their flat companions. I'm putting my vote for the baby walker.
But the 12 x 24 bricks have a more standard bottom, and work like normal bricks.
All sorts of things from sprues are smaller, but they're really parts of parts.
The smallest on a chemistry scale was the minifig coin (smaller than the "lever handle" in a small lever). But, obviously, not everything was tried. A few other candidates at the time (that weren't measured) were crazy things like the center element from a Technic universal joint.