The Highest Dice

Perl 6, 24 23 bytes

-1 byte thanks to nwellnhof

{.put;.[]X~(6...1)}...*

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Outputs the sequence infinitely separated by spaces/newlines. Or, for a few more bytes we can have a lazy infinite list we can index into instead.

Perl 6, 27 bytes

{flat {@=.[]X~(6...1)}...*}

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Explanation:

{                         }    # Anonymous code block
 flat                          # Return the flattened
                      ...*       # Infinite sequence
      {              }             # Defined as
         .[]                       # The previous element arrayified
            X~                     # Each concatenated with
              (6...1)              # All of 6 to 1
       @=                          # Arrayified

Python 2, 39 38 34 bytes

f=lambda n:n and-n%6+1+f(~-n/6)*10

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Outputs 1-indexed number

Bash, 31 bytes

f()(x+={6..1};eval echo $x;f);f

TIO

update from comments, the n'th value 1-indexed, +GNU tools + perl, 64 bytes, 7 bytes saved thanks to @manatwork

dc<<<6o$1p|perl -pe 's/(.)0/($1-1).6/e?redo:s/0//'|tr 1-6 654321

64 bytes