Really, really, really hard sequence

... in which you have nothing to do with math
... except maybe with some counting and this and that
... but no calculations, no, definitely no.

$1, 0, 8, 62, 90, 68, 83, 89, 88, 057, 619, 028, 693, 069, 068, 985, 086, 988, 871, 820, 838, 865, 890, 868, 887, ?, ?, ?, ...$

Question:

What are the next elements?

Hint 1:

4 is missing for reasons.

Hint 2:

4 is too much.

Hint 3:

4 is somehow ambiguous.

The next numbers are:

$889, 888, 0123$

Because

they are ternary representations of the natural numbers starting at $0$, where the trits are the number of holes in the base $10$ digits
The next ternary numbers to represent are $221, 222, 1000$.
You are cycling through $(0,6,9)$ for those with one hole,
through $(1,2,3,5,7)$ for those with no holes,
with just $8$ for two holes.

$4$ has a hole so it could be used but you decided not to do so because it may also be written without one
($0$ could be written with a slash through it too, giving it two holes).

• Well... half way done. Probably because of my fault with exchanged elements (sorry for that). Answer is unambiguous, so... Well done so far :) – kamenf May 22 '16 at 1:51
• @kamenf I updated it to use the rest of the methodology I believe – Jonathan Allan May 22 '16 at 1:59
• Now it is perfect. And about 0... well, when I was at school there was no such thing as slashed zero, so I totally forgot that case :-D – kamenf May 22 '16 at 2:04
• Yeah, it does not matter really, just thought I'd pop that in there - you are writing the numbers after all :) – Jonathan Allan May 22 '16 at 2:07
• @warspyking I don't think any conversion (ether from ternary to decimal or vice-versa) is needed to solve this puzzle. Even to present them as text. Just digits. :) – kamenf May 22 '16 at 19:36