Sonic The Hedgehog has a pretty cool one. If you press A B & C simultaneously, Sonic starts doing stupid actions, and of you let them go, he’ll try to do his usual routine, but hilariously messed up.
Metroid fusion also has the “tutorial” demo, in which it teaches you abilities and secret locations of you decide to watch it. Super Pitfall has the infamous clear game demo “glitch” and a couple of NES games have the “this game is so hard, even the demo player can’t survive for very long” demos.
My favorite and more nostalgic one has to be Super Mario World. Not only it shows a super secret stage, it also serves as the main menu (and the music really catches what that game is all about).