Life Force-NES
A space shooter that is impossible to beat this game without Konami's 30 lives code (Up, up, down, down...). There are many obstacles that can only be passed through trial-and-error.
The Legend of Zelda-NES
The game is easy until Wizzrobes and Darknuts become a common nuisance. (**** enemies taking three hearts away from you every hit)
Solomon's Key-NES
The puzzles in this game are unbelievable. Creating blocks, trapping enemies, and finding the end of the levels are what make this game challenging. I think there are 50 levels. I couldn't get past level 11.
I realize these are all NES games, but this was gaming at its best people.
Note: I also second OoT-Master Quest