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I second Lemmings although there are lots of hideous deaths and you can blow them all up!
Other ideas:
Speedball 1/2
Lotus 1/2/3
Monkey Island 1/2 (I think there is no shooting)
Sensible Soccer/World of Soccer
Pinball Dreams/Fantasies etc.
Jimmy Whites's Whirlwind Snooker
Moonstone
Bubble Bobble/Rainbow Islands
And many more...
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