null
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000...?
Or maybe,
0101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101...?
Or maybe with more complexity,
000102030405060708090a0b0a090807060504030201000102030405060708090a0b0a090807060504030201...?
Of course, the repeating character of null is utterly destroyed when the process is repeated, so perhaps,
0,10,1110,11110,10010,111001110,11110010110,...
It seems to have plenty of imperfect repeating character, but repetition is far more likely with only two symbols or four:
0,10,1110,3110,132110,1113122110,311311222110,...
There seems to be far less repetition in base 4 than in base 2 (and overall answer: stumped for moment).
ADDITION:
The OEIS states that the sequence is eventually periodic with period 2 or 3.
0,
10,
1110,
3110,
132110,
1113122110,
311311222110,
13211321322110,
1113122113121113222110,
31131122211311123113322110
132113213221133112132123222110,
11131221131211132221232112111312111213322110,
31131122211311123113321112131221123113111231121123222110,
1321132132111331121321231231121113112221121321133112132112211213322110
New attempt:
2110,122110,11222110,21322110,121113222110,...
But still, nothing that repeats itself 100% and not more.