Create a square of increasing size by replicating the initial code

Pyth, 2 bytes

5

Try it online! Also Try it doubled, tripled!

How does that work?

\n is the command that prints its argument with a trailing newline, while returning it simultaneously. So, each time you make an insertion, you turn the integer literal 5 into a number containing N copies of 5 concatenated, and the leading newlines basically make sure it's printed the appropriate number of times, thus keeping it square.

JavaScript (ES6), 42 32 30 bytes

s=[this.s]+0;  console.log(s);

Second iteration:

s=[this.s]+0;  s=[this.s]+0;  console.log(s);console.log(s);

This works by appending a 0 to s each time the first half of the code is run, and printing s itself each time the second half is run. Takes advantage of four quirks of JavaScript:

1. The current environment can be referred to with this. This allows us to do this.s in place of s.
2. When accessing a property that has not been defined on an object, instead of throwing an error, JavaScript returns undefined.
3. An array plus a number returns a string. [1,2,3] + 4 === "1,2,34"
4. When stringifying an array, undefined is converted to the empty string, which means that [undefined] + 0 === "0".

Put together, this means that we can express the first half (generating a string of zeroes) in just 13 bytes. If using alert instead of console.log is allowed, we can save 4 more bytes by shortening the second half.

05AB1E, 2 bytes

5=

Try it online!

Port of my Pyth answer.