Is it possible to build 3 or more concentric circles / nested circles / telescoping pieces? There are round 2x2 pieces.
There are round 4x4 pieces with round 2x2 holes, the 2x2 can slide/rotate inside of it.
There are "macaroni" pieces to make an 8x8 circle with a 6x6 hole.
But are there no 6x6 rounds with a 4x4 hole?

I've also seen sets use curved slopes, curved inverted slopes, and arches to make circles or nearly-perfect-circles. Are there ways to make nested circles from those pieces?
There is only one piece that fits all your criteria: 41823. Placing four of those pieces together will create a 6x6 circle with a 4x4 interior gap. Albeit, it is a decorative fence piece and it's two studs tall, so it might not work for what you are trying to accomplish.