Let's call a positive integer N self-indulgent of degree K>2 if for every positive integer k<K the following is true:
More than half of the first k multiples N,2N,...,kN of N contain with multiplicity all the digits of N. So, if, for example, the digit 4 occurs three times in the decimal representation of N, then it has to occur three or more times in that of the multiple.
Do self-indulgent numbers exist? If yes, can you give one of degree at least 10?
Attribution: Mine I think but wouldn't be too surprised if to coin a phrase I independently rediscovered it.