When I was a kid I had an atari st. i really loved this game called 'black lamp'. i guess... you did shoot things... but it was a cartoon style game that was really about finding all the different coloured lamps and putting them in the boxes. there was a 3-d pacman that was pretty good too :-)

do you remember the hand held games that used to have one game per unit? tennis was a big favourite.

frogger is ringing a bell too... where you get the frog over the road between the cars then over the river on the logs.

um...

i'm a bit of a minesweep nut but i guess that is really very nerdy...

and spider solitare and freecell still keep me amused at times...

aside from that i guess i like the first person role playing games (such as heroes of might and magic and temple of elemental evil etc etc). i really love different kinds of spells and saving up for the extra cool armour... but yeah i guess there is fighting in that...

x-box... i used to play the harry potter games which weren't too bad.

i've never played myst but there is meant to be a sequal to that too that is pretty good. not violent, more puzzle type games...

not too sure about the non-violent aspect... (sorry)

project gotham?