(I started working on this problem after trying to get any "interesting" pattern out of the number that Gowers randomly wrote while answering:What is realistic mathematics?.)
The number was 123871205412470874297947938271423698765734564756028492656.
Take any number, for instance:
3484756955 (in the preceding number there are three 0's, four 1's,..., and finally five 9's)
0001231111 (in the preceding number there are no 0's, no 1's,..., and finally one 9)
.... clearly the list won't end as there will always be some no 0's or 1's or 2's, etc.
A variant of this sequence is discussed here(Conway's look-and-say sequence).
This a fairly simple obsevation, so is there a literature about such sequences from which I can learn more?