Here is just a list of stuff the player might be able to do after finishing the game other than do speed runs and collect all the Large Squares. Here are my first few ideas:
-endless enemy spawner that gets increasingly difficult as time passes.
-secret super hard level/group of levels that test even the most skilled players
-mini games!
-optional bosses that reward you with sweet loot when you beat em
-secrets to find in all the levels that reveal more about the hero's past
-insane mode. Like the current hard mode, but all enemies have 10 times as much HP and the player only ever has one health at any given time! Enemies also gain one special ability to use against the player (eg. heal, dash at the player really fast, turn invisible [you can still see their outline], etc.)
Feel free to add your own suggestions