I was scared out of my mind by the enemies on the PS1 game, Spyro 2: Ripto's Rage (1999). I would always just stand somewhere out of their NPC detection range and watch them while I would screw around walking in circles or flying over obstacles wherever I was. If I accidentally landed somewhere causing an enemy to react, I would immediately struggle to get back to my safe haven. If an enemy managed to deal damage to me, my heart would be racing beyond 120bpm and I would usually scramble to pause the game and shut off the console. I don't remember when I conquered my fear (9 or 10 years old?), but I've beaten all the levels I've played.
On driving games with police cars where the cops would either pull you over, arrest you, or wreck your car, I would always use cheats to disable pursuits and/or remove their cars altogether (exception on the 1994-6 PS1 game, Road & Track's Need For Speed, since the police were usually of no bother). On Driver (as well as Driver 2), I refused to complete story mode because it was too easy without cops and too "hard" with cops.
On Sled Storm (PS1) and Road & Track's Need For Speed (PS1), I was scared of the pixellated 2D structures used to simulate wildlife (bears and moose respectively) because I thought they would run into the road causing an accident and/or chase me and try to get me.
I was scared of playing FPS games involving firearms (and banned from playing them anyway) until I was 11 or 12 years old because I thought the blood would be scary (only when very little, after watching my dad play them this became obsolete) and that the guns would be too loud in comparison to the volume. I thought I would put it on the wrong volume and then firing the gun would be insanely loud.