formation of numbers - New Mathematics forever

^{This has been (renamed and) refocused on boboquack’s unforeseen solution, which deserves its own puzzle.}

^{Right when you thought the good old days of New Mathematics were safely dusted, along comes an unending lesson on similarities of numbers in different bases. For this  lesson  puzzle, 100₈ means the digits 100 in base 8, which equals the familiar 64 (in base 10), which itself is represented as 64₁₀.}

   Just when is 100 the same as 64?   When isn’t it?!   Just look:

          100₈    =  64₁₀ ?   ✓
          100₁₀  =  64₁₆ ?   ✓
          100₁₆  =  64₄₂ ?   ✓   Could this go on forever?

   That used 6 different digits in all — 0, 1, 2, 4, 6, 8 — and may be summarißed as...

           K  _W  =  L _X
           K _X   =  L _Y
           K _Y   =  L _Z

   ...where   K = 100,   L = 64,   W = 8,   X = 10,   Y = 16   and   Z = 42.

   And no, this pattern does not continue forever, not even past L _Z ( 64₄₂ ).

But wait,  other patterns can indeed go on forever.   And with fewer different digits.

         M _P   =  N _Q
         M _Q  =  N _R
         M _R  =  N _S
         M _S   =  N _T
           .
           .
           .

What pattern of M, N, P, Q, R, S, T,  .  .  .   uses the fewest different digits?

How can that be generalized to other numbers of different digits?

( M > N   and   P < Q < R < S < T <   · · · )

To quote professor Tom Lehrer from New Math _YouTube, “Hooray for New Math!”

Answer

A little touch-up to the previous answer gives a proper solution:

I doubt you can do it with just one digit, but feel free to prove me wrong!

$$\begin{array}{lcl}100_{10}&=&10_{100}\\100_{100}&=&10_{10000}\\100_{10000}&=&10_{100000000}\\100_{100000000}&=&10_{10000000000000000}\\&\vdots&\end{array}$$
$$\begin{array}{lcl}200_{10}&=&20_{100}\\200_{100}&=&20_{10000}\\200_{10000}&=&20_{100000000}\\200_{100000000}&=&20_{10000000000000000}\\&\vdots&\end{array}$$

$$\begin{array}{lcl}203_{10}&=&23_{100}\\203_{100}&=&23_{10000}\\203_{10000}&=&23_{100000000}\\203_{100000000}&=&23_{10000000000000000}\\&\vdots&\end{array}$$
Note: the pattern really begins here: $$\begin{array}{lcl}40203_{10}&=&423_{100}\\40203_{100}&=&423_{10000}\\40203_{10000}&=&423_{100000000}\\40203_{100000000}&=&423_{10000000000000000}\\&\vdots&\end{array}$$
$$\begin{array}{lcl}5040203_{10}&=&5423_{100}\\5040203_{100}&=&5423_{10000}\\5040203_{10000}&=&5423_{100000000}\\5040203_{100000000}&=&5423_{10000000000000000}\\&\vdots&\end{array}$$
$$\begin{array}{lcl}605040203_{10}&=&65423_{100}\\605040203_{100}&=&65423_{10000}\\605040203_{10000}&=&65423_{100000000}\\605040203_{100000000}&=&65423_{10000000000000000}\\&\vdots&\end{array}$$
$$\begin{array}{c}\vdots\end{array}$$