Join N copies of a snippet to produce N^2 characters
TECO, 4 bytes
V1\V

V prints the contents of the current line in the text buffer. 1\ inserts the string representation of the number 1 at the current position.
So on the Nth iteration of the program, the first V will output N - 1 copies of the character 1, then add another 1 to the text, then output N 1s.