Stax, 20 19 18 16 bytes

ù↔‼i,ÑΓæ☺=╘‼æ↕4╝

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Explanation:

c%R:BX%mYx%{y|Mvx@m Full program, implicit input
c%                  Length of input
  R                 1-based range
   :B               Repeat each element according to the range ("123" -> "122333")
     X              Save to X register
      %             Length
       m            Map over 1-based range:
        Y             Save index to Y register
         x%           Push length of X register
           {      m   Map over 1-based range:
            y|M         Maximum of both indices
               v        1-based -> 0-based (decrement)
                x@      Index into X register
                      Implicit output with newline

Stax, 12 bytes

ü°√¿«│⌠º₧@\τ

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Using this algorithm.

Explanation:

c%R:BXm]i*xit+ Full program, implicit input
c%             Length of input
  R            1-based range
   :B          Repeat each element according to the range ("123" -> "122333")
     X         Save to X register
      m        Map:
       ]         Character -> string
        i*       Repeat by iteration index
          xit    Trim first <iteration index> elements from X
             +   Concatenate
                 Implicit output with newline

Stax, 20 19 18 16 bytes

ù↔‼i,ÑΓæ☺=╘‼æ↕4╝

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Explanation:

c%R:BX%mYx%{y|Mvx@m Full program, implicit input
c%                  Length of input
  R                 1-based range
   :B               Repeat each element according to the range ("123" -> "122333")
     X              Save to X register
      %             Length
       m            Map over 1-based range:
        Y             Save index to Y register
         x%           Push length of X register
           {      m   Map over 1-based range:
            y|M         Maximum of both indices
               v        1-based -> 0-based (decrement)
                x@      Index into X register
                      Implicit output with newline

Stax, 20 19 18 bytes

ü├¶∟↓-[jvªjGM+J☺^)

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Stax, 20 19 18 16 bytes

ù↔‼i,ÑΓæ☺=╘‼æ↕4╝

Run and debug it

Explanation:

c%R:BX%mYx%{y|Mvx@m Full program, implicit input
c%                  Length of input
  R                 1-based range
   :B               Repeat each element according to the range ("123" -> "122333")
     X              Save to X register
      %             Length
       m            Map over 1-based range:
        Y             Save index to Y register
         x%           Push length of X register
           {      m   Map over 1-based range:
            y|M         Maximum of both indices
               v        1-based -> 0-based (decrement)
                x@      Index into X register
                      Implicit output with newline

Stax, 20 19 bytes

╛▲G≥ôÿ☻*_:ÉZ╪;Z4↕‼╝

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Stax, 20 19 18 bytes

ü├¶∟↓-[jvªjGM+J☺^)

Run and debug it