Hardest games I've played and can't beat:
Battletoads Level 1
Zelda II Great Palace
Abadox Level 2
Tom & Jerry Level 1-1
AOBB (Adventures of bayou Billy) LEVEL 1 main game
Not hard games and can't beat:
Finding nemo the game level 6
I beat Sonic & The secret rings but can't do the perfect challenges
Super Mario bros warpless, no continues
SMB2TLL world 6 luigi
S3&K Doomsday zone
Startropics 3rd dungeon
The list continues on and on...
Tell me the hard games that you could never beat and maybe I'll help you, it'll depend on the game.
There are many difficult games here and there that were unbeatable in your age.

Stevie 764 : wow this is a nice thread you  made good job!  You dont have to offer any games tips on them .

-Double Dragon one with and without game genie
-Double Dragon 3 first level is way to hard to beat even with infinite health cheat on the game genie
-FX zero on the gamecube
-Several 2D fighting games such as Power Instant:Matrimilee,Rage of the Dragons,Vampire Hunter arkstalker Revenge, Fatal Fury one Street Fighter one ,Tatsunoko vs Capcom  Psychic Force 2012 and Real Bout Fatal Fury.
-Professor Layton and the Curios Village yeah I'm pretty dumb enough  on solving  many puzzles in that game.
-Super Mario bros 2  Japan  had a good reason not to release that game in the U.S ITS HARD!!!     
 - Ninja Gaiden 3   They  no longer offer infinite continues like in the first two
- R Type 3
Mega Man Neo   on vizzed I wonder who's the tard that didn't even input in the power charge and powerslide in the game.
So the most hardest games  around are the oldest 2D classic games .

I have played Ninja Gaiden 2 and rage-quit during the final boss battle.  I still haven't beaten it.  

nintendoninja64 : OH MY GOD!!! A 5 month thread bump? Makes me wish it was a active thread.

Differences to my game chart:

SMB2TLL fully complete w/ luigi.
I will try Doomsday Zone on Sonic 3 & Knuckles soon.
I will try to do a warpless Super Mario Bros game without getting a single Game Over soon.

Ninja Gaiden NES, got to 6-2 legit, but could not get passed 6-2 without a game genie code.
I Wanna Be the Guy, Because......(do i really need to explain this)
Double Dragon III NES, Level 2. Yeah, I got past the first level on my own. With a bit of practice, I could complete the game.
Super Turrican SNES, Couldn't get past the level with the moving alien trains, had to use the level skip code =|

That's pretty much it. I'm not including Superman 64 for the N64 because I am going to complete it (got to level 6/7, lost count). There are others, but those are minorities on terrible games.

Sonic 3 and knuckles doomsday zone, smb 8-3