Compact packings of the plane with three sizes of discs: https://arxiv.org/abs/1810.02231, Thomas Fernique, Amir Hashemi, and Olga Sizova
Here, "compact packing" means interior-disjoint disks forming only 3-sided gaps. The circle packing theorem constructs these for any finite maximal planar graph, with little control over disk size. Instead this paper seeks packings of the whole plane by infinitely many disks, with few sizes. 9 pairs of sizes and 164 triples work. Here's one from Fig.3 of the paper.
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