Runic Enchantments, 6 bytes
"'<S@>
This one was weird. All I had to do was remove a
~
from the original quine found by Jo King.
Every additional run appends another
<
to the end, e.g.:
"'<S@><<<<<<<<<
All of which do nothing.
Direct copy of this answer on a related challenge. It just so happened to already grow by 1 byte every iteration (strong argument for this challenge being a duplicate of that one or vice versa).