I honestly try to beat every game I like. But I tend to give up after X amount of tries. Either a game sends me back too far to the last checkpoint for me to want to subject myself to pain over and over, or I hit a roadblock and just cant continue. Here are a few games where I just had to say "enough is enough"

Road Rash 2 - love the game but it just becomes unforgiving at level 4. Any time I get some speed, I cant keep it on the road and I either crash and get busted, or totaled. And you need to drive fast at these stages.

Castlevania - I made it to Frankenstein I believe. It didnt seem terribly difficult but in the leadup to the battle I just lose a ton of health

Circle of the Moon - I made it to Camilla and Im surprised I made it this far. The boss battles in this game are tough but this one might just have me say "f*** it"

Toki - I got to the last boss a few times and just couldnt beat him. I dont want to have to replay the entire game every time I want to face him

Joe and Mac 2 - facing every boss before the final boss is just sadistic. I can usually fare pretty well but then I can only beat Gorks first form.

Metroid 2 - I got lost in this game. No idea where I was supposed to go. My battery died and I lost my save.

Donkey Kong Land - I missed some of the bonuses, no idea where they are. But I sure as hell tried.

Contra - I cant remember how far I got and its been a while since I played it but I dont think I beat it. This one seems doable if I go back and try to replay it.

Ghosts and Ghouls - its a tough game but I like it so may give it another shot

Battletoads - I can get to the boss of Intruder Excluder. It doesnt seem terribly hard but at that point Ive lost so many lives, especially from the fight with Blag. Some day Ill go back and try to beat it.

What amazes me is some games I expected to be hard, I was able to easily finish- Gargoyles Quest 2, Monster Party come to mind, but commonly beaten popular games Ive struggled with.