One approach I'm thinking that could work is a solid core, and then a fibrous material all around. Something that resists stretching. Build it in 100m sections, each section has the fibre material wrapped in various up down/cross cross patterns and always start at the very bottom so each one is encapsulated and the outer layer stretches from bottom to very top. My train of thought is the fibrous material would resist stretching, keeping the tower straight, and the core would resist crushing. The fibre would also resist the core from stretching as it tries to crush itself. I wonder what would be the limit of this using carbon fibre and steel as those are the two materials that come to mind that we have access to in reasonable amounts. You would also want to start the build process very deep under ground, like 1km or so, then backfill with concrete. In the end the tower is practically one big item vs a bunch bolted together.
I don't know if this could actually reach 1km, let alone 100 though...