Dynamite Headdy/YMMV

Awesome Music: Tons, but especially the epic final boss theme, and the cast theme. Nasty Gatekeeper's theme makes for a brilliant Oh Crap too. The tribute album Secret Bonus Point only makes it better.
That One Boss: Treasure does everything to the Nasty Gatekeeper short of putting a sign over her head saying "this is That One Boss". The introduction, the music, and of course the difficulty (at least until you get the hang of it). then again, a little later on you get Twin Freaks which doesn't have the intro or music, but sure has the difficulty. Compared to them, the Final Boss is a cakewalk once you know how to fight him. Even before that there is, in the Unexpected Shmup Level, Baby Face, who puts you through laser hell while repeatedly reaching up to grab you with a hideous hand.
That One Level: The last little part of Fun Forgiven has a bunch of Hangmans flipping in and out of usability. Unless you know what you're doing, this part is very annoying to get through.
Unwinnable By Insanity: On the first real level, you have to catch a nearby platform to be able to climb a wall. As seen here, if you bring the platform too far away from the ledge or the other platform, you won't be able to continue. You can't move the platform again, and, provided you killed the only enemy around, you can't kill yourself, forcing you to reset the game.