Motivated by sol-gel transitions, David Aldous (2000) introduced and analysed a fascinating dynamic percolation model on a tree where clusters stop growing (`freeze') as soon as they become infinite.
In this talk I will discuss recent and ongoing work (with Demeter Kiss and Pierre Nolin) on processes of similar flavour on planar lattices. We focus on the question whether the giant (i.e. `frozen') clusters occupy a negligible, moderate or very large fraction of space. A related question is whether microscopic `trapped' regions occur. It turns out that the behaviour is very different from that for trees. Accurate results for near-critical percolation play an important role in the analysis.