I think I found a concept I like. I’m currently calling it “Pillar”, as in a pillar/tower, and caterpillar. Here’s some concept art.

A character composed of blocks, that acquires new blocks as he moves. The controls I haven’t figured out entirely, but at the very least I think it’ll be based around the idea of picking a block on the character as a pivot point, and rotating him. Probably constanly moving, walking in to walls, until told to turn around (or automatic after a few seconds).
As blocks are collected, they become part of him, evolving in to Tetris like shapes and beyond. The rotating mechanic is your ability to get to new heights, and cross gaps. Physics wise, I think it’ll be entirely axis aligned, since it’ll be far too easy to make shapes that couldn’t normally keep balance. Legs would sprout wherever needed, purely for look cohesiveness. Face always moves to the front middle of the shape. Perhaps tails and other things sprout if a shape deems it.
My only concern at this point is trying to figure out, level design wise, why you wouldn’t want to always create a long piece.