If something puzzles you, don't be afraid to start drawing yourself a diagram. O(n^2) vs. O(n) amortized for naive string concatenation vs. growing a buffer by doubling:

  x
  xx
  xxx
  xxxx
  xxxxx
  xxxxxx
  xxxxxxx
  xxxxxxxx

  x
  xx
  xxxx
  xxxxxxxx

This can be a good way of getting an intuitive feel for what's going on. Note in the 2nd case, you can stack all of 1st two rows representing cost into the 3rd row. In fact, you can always stack the execution costs into row (n-1) and never exceed the size of row n.