echo | sed '
  s/.*/1/
  :a
    s/$/c/
    h
    s/[05]c.*/&buzz/

    s/[0369]/333/g; s/[258]/33/g; s/[148]/3/g
    s/^\(333\)\+c/&fizz/

    /z/{ s/.*c//; p; g; bb }
    g; s/c.*//; p; g

    :b
      s/c/c0123456789cx/
      s/\(.\)c.*\1\(.\).*x/\2/
      s/c/c0/; s/^c/1/;
    /c/bb
  ba'

Or the short form:

echo | sed 's/.*/1/;:a;s/$/c/;h;s/[05]c.*/&buzz/;s/[0369]/333/g;s/[258]/33/g;s/[148]/3/g;s/^\(333\)\+c/&fizz/;/z/{s/.*c//;p;g;bb};g;s/c.*//;p;g;:b;s/c/c0123456789cx/;s/\(.\)c.*\1\(.\).*x/\2/;s/c/c0/;s/^c/1/;/c/bb;ba'

echo | sed '
  s/.*/1/
  :a
    s/$/c/
    h
    s/[05]c.*/&buzz/

    s/[0369]/333/g; s/[258]/33/g; s/[148]/3/g
    s/^\(333\)\+c/&fizz/

    /z/{ s/.*c//; p; g; bb }
    g; s/c.*//; p; g

    :b
      s/c/c0123456789cx/
      s/\(.\)c.*\1\(.\).*x/\2/
      s/c/c0/; s/^c/1/;
    /c/bb
  ba'

Or the short form:

echo | sed 's/.*/1/;:a;s/$/c/;h;s/[05]c.*/&buzz/;s/[0369]/333/g;s/[258]/33/g;s/[148]/3/g;s/^\(333\)\+c/&fizz/;/z/{s/.*c//;p;g;bb};g;s/c.*//;p;g;:b;s/c/c0123456789cx/;s/\(.\)c.*\1\(.\).*x/\2/;s/c/c0/;s/^c/1/;/c/bb;ba'
[cancel] password:  [private]