Solution:

Starting from $n^2$ tiles, one operation removes the $n$ perfect squares, leaving $n^2-n.$ The next operation removes $n-1$ perfect squares, leaving $n^2-n-(n-1)=(n-1)^2.$

So every two operations reduce $n^2$ to $(n-1)^2.$ Going from $10^2=100$ down to $1^2=1$ takes $2(10-1)=18$ operations.

Thus, the correct answer is C.