The point growth fractal (coral) in the Euclidean plane (or space) is a battle between trees, which would like to branch and grow exponentially, and space, which grows only polynomially with radius.
Make the battle more severe by growing it on a (say) horizontal strip. Cylinder also possible. Mouse over a point to highlight the descendants. Pick a vertical line of pixels and it highlights the most recent common ancestor.
On the other end of the spectrum, make area or volume increase exponentially also: hyperbolic space. Conveniently, any regular polygon (pentagon or more) tiles (some) hyperbolic plane, so can be used as a pixel. A few possibilities for hyperbolic 3-space.
Even more extreme, grow it directly on an empty infinite binary tree. The point wanders among empty siblings or cousins, parent, or children until it occupies node with a filled parent: there are 5 possible directions of a step. This will likely not be interesting, because there seems to be no possibility of crowding out.
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