Generate n digits of Gijswijt's sequence
Pyth, 25 22 21 bytes

t_u+eSmtfxG*Td1._GGQN


OP confirmed that we only need to handle single-digit numbers. This allowed to store the list as a string of digits. -> Saved 3 bytes

Try it online: Demonstration

Explanation:

t_u+...GQN      implicit: Q = input number
         N      start with the string G = '"'
  u     Q       do the following Q times:
    ...            generate the next number
   +   G           and prepend it to G
 _              print reversed string at the end
t               remove the first char (the '"')


And here's how I generate the next number:

eSmtfxG*Td1._G
           ._G    generate all prefixes of G
  m               map each prefix d to:
    f     1          find the first number T &gt;= 1, so that:
       *Td              d repeated T times
     xG                 occurs at the beginning of G
 S                  sort all numbers
e                   take the last one (maximum)   


21 bytes with lists

_u+eSmhhrcGd8SlGGtQ]1


Try it online: Demonstration

Uses the same ideas of Martin and Peter. At each step I split the string into pieces of length 1, pieces of length 2, ... Then I range-length-encode them and use the maximal first run as next number. 

20 bytes with strings

t_u+eSmhhrcGd8SlGGQN


Try it online: Demonstration

Combines the ideas of the two codes above.