1235813941213533488671216141383377554...

Is there a limit to the number of times in a row a digit can appear?

The length-9 string of 1's (111111111) at position 20,560 was traced backto 1010101010 at 16,011. That can be traced back to 555555 at 12,439, and that in turn to 1414141 at 9,689. Going back farther:

77775 at 7541
161614 at 5874

At this point it traces back to 8886 at 4560.  But subsequent 161614's (and ultimately 111111111's, etc.) are traceable to different sequences, such as 7977.

But, continuing the first 111111111's path, the 8886 at 4560 is traceable to

71715 at 3544
61078 at 2753
15526 at 2131
014115 at 1649
37741 at 1281
125229 at 993

etc.

Thus happenstance leads to increased order (going up the list and outward on the string).