2 of 9 added 126 characters in body

Java 8, 77 93 bytes

int n=1111;int m=1;while(x>m++){n=Integer.parseInt((n*n+"").substring(0,4));}

x->{int n=1111;int m=1;while(x>m++){n=Integer.parseInt((n*n+"").substring(0,4))‌​;}return n;}

Each repetition makes n the first 4 characters of n*n.

Try Java online!