The key detail here is that the new pieces don’t get added to the string in place, one by one. What actually happens is that each time around the loop, the previous contents of the line variable become dead - a whole new string is allocated to contain the original piece plus the new part at the end. Since the string gets longer with increasing values of i, the amount of heap space being consumed also increases and so it is easy to use up hundreds of bytes of free heap space each time this function is called. If you need to concatenate many strings together then a much better option is the Mono library’s System.Text.StringBuilder class.
However, even repeated concatenation won’t cause too much trouble unless it is called frequently, and in Unity that usually implies the frame update. Something like: