You can design a number. Just take all finite digit strings in order of length and numerical order: 0.123456789 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 ... 99 000 001 002 ... 999 0000 0001 ...
obviously it contains every finite digit string in base 10. I can't prove the digits are uniformly distributed in every base - you'd have to be more clever but you see the idea.
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You can design a number. Just take all finite digit strings in order of length and numerical order: 0.123456789 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 ... 99 000 001 002 ... 999 0000 0001 ...
obviously it contains every finite digit string in base 10. I can't prove the digits are uniformly distributed in every base - you'd have to be more clever but you see the idea.