How many cardboard digits do I need?
Jelly, 9 bytes

‘ḶDœ|/ḟ9L


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How it works

‘ḶDœ|/ḟ9L
‘Ḷ         [0,1,...,n]
  D        convert each to list of its digits
   œ|/     fold by multiset union
      ḟ9   remove 9
        L  length