What is the sum of the digits of the number $10^{25}-25$?

Try small cases first: $10^3-25=975$ and $10^4-25=9975$, so the borrowing turns the leading digits into 9s and leaves 75 at the end. Since $10^{25}-25$ has 25 digits and the last two are 7 and 5, the first 23 are all 9. The digit sum is $23\times 9+7+5=207+12=219$. Stopping at 207 counts only the nines, and $225=25\times 9$ wrongly assumes all 25 digits are 9.