The definition for ordinal-time cellular automata is simple:
Generation 0 is just the initial state, as it always is.
Generation n+1 is the cellular automaton applied to generation n, as it always is.
Generation A, with A being a limit ordinal, makes any cell that is alive infinitely often just before generation A be alive.
Here is a video of the blinker’s evolution in infinite-time Conway’s Game of Life up to generation ω+8:
The blinker’s evolution actually continues past the generations shown above, and it eventually stabilizes at generation ω5+25.
Other common oscillators stabilize way earlier: a toad stabilizes into a block at generation ω+3, a beacon is already stable (even though it’s oscillating), a clock stabilizes into a fleet at generation ω+8, and the n-barberpole stabilizes into a longn ship at generation ω, and the pulsar stabilizes into eight blocks at generation ω+22.
No pattern has a known stabilization point past ω22. To be clear, in this post, stabilization refers to the first generation that doesn’t stop appearing at some point.
The only known online appearances of this are https://googology.fandom.com/wiki/User_blog:LittlePeng9/Infinite_time_game_of_life, which defines it, a thread in the #misc-ca channel of the ConwayLife Lounge Discord server that has links to all of the other examples in this list, https://scratch.mit.edu/projects/1267474269/, which simulates it on random initial states for generations less than ω3 (and lets you move through the generations freely!), and this blog post.