I've completed quite a few games but now a days I do cheat <g>.. however back in the day I did:
Another World - easy
Flashback - easy
Dungeon Master - took weeks and was solid in places - not knowing about the altar of rebirth (obvious now) at the time was a nightmare trying not to die.
Turrican 2 - hard - very hard
Bubble Bobble - absolutely solid - first time I completed it, it said bring a friend.. arrgh!
Rainbow Islands - again hard... lots of practice needed
Speedball 2 - pretty easy once you bought a few players..
Supercars / 2 - pretty easy - rockets rock!
Monkey Island - hard in places but very good fun!
Darkseed - absolutely solid having to be in the right place at the right time
Elite 2 - does getting to Elite mean I've completed it? <G>
Xenon 2 - hard in places.. very hard..
SWIV - as above
X-Out - pretty easy
Turrican - pretty easy if I remember right
Last Ninja 2 - fecken hard - especailly some of the jumps
And lots more which will come to me soon!