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I would like to do something like CellularAutomaton[10, {1, 0, 0, 0, 0}, 2^5], but instead of running for 2^5 steps, it should stop as soon as a configuration has been seen twice, and report the number of such steps taken. Is it possible to do this?

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5
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You can use the one-step functional version with NestWhileList:

NestWhileList[CellularAutomaton[30, #] &, {1, 0, 0, 0, 0}, UnsameQ, All]

{{1, 0, 0, 0, 0}, {1, 1, 0, 0, 1}, {0, 0, 1, 1, 1}, {1, 1, 1, 0, 0}, {1, 0, 0, 1, 1}, {0, 1, 1, 1, 0}, {1, 1, 0, 0, 1}}

For the number of steps taken until a configuration repeats:

Length @ %

7

Note: In general, CellularAutomaton[rule, init, n] can also be obtained by nesting the one-step functional version using NestList[CellularAutomaton[rule, #]&, init, n]:

CellularAutomaton[30, {0, 0, 1, 0, 0}, 2^5] == 
  NestList[CellularAutomaton[30, #] &, {0, 0, 1, 0, 0}, 2^5]

True

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    $\begingroup$ One could use UnsameQ instead of DuplicateFreeQ @ {##} & in the third argument. $\endgroup$ – J. M. is away♦ Oct 17 '18 at 1:20
  • $\begingroup$ @J.M. Right; much better. Thank you. $\endgroup$ – kglr Oct 17 '18 at 2:26
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Clear[f]

f[state_] := f[f[state] = CellularAutomaton[30, state]]

{f[{1, 0, 0, 0, 0}], Length@DownValues[f]}

{{0, 0, 1, 1, 1}, 7}

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