I'm amazed so many people posted "Blaster Master".
Loved that game when I was a kid and beat it quite a few times, though I'll grant the lack of a save function/limited continues was a bitch.
Admittedly, I probably only managed to beat it because I had the Max controller with the turbo button (can we say "turbo grenades?"), not to mention the cheap-shot trick you could do to beat some of the bosses (grenade them, then pause the game for about a minute; the boss takes damage from your grenade while the game is paused). The controller also helped immensely in maneuvering at the end against the final boss.
Fun game, though. Loved the upgrades you got to your tank after each boss.
As far as most frustrating game ever: I'll go with Super Mario Sunshine. Uuuuuuggggghhhhh. Sheer stubbornness was the only thing that made me finish that game, despite the numerous levels with the obstacle courses, not to mention the stupid "clean up the beach in 3 minutes" level.
Edit: Any of the SimCity games I ever played. Maybe I'm just stupid, but I never got the hang of making money in any of them. Always ended up starting over.