I stumbled on these a few years ago, and keep going back to them. If you're looking for something to do on the Web with your infant or toddler, try out Kneebouncers. You play the games by hitting any key on the keyboard (so easy, an adult can do it).
For older kids, (and adults who still have some hand-eye coordination left), try the beautifully-designed Orisinal: Morning Sunshine Games by Ferry Halim.
The illustrations and subjects are sweet, and many of them don't involve hitting or shooting. (The ones that do seem innocent and are not bloody or gory.) I like the ladybug one ("Bugs"), but apparently the bee game is the most popular. A bonus: the music is tinkly and calming -- no blasters, driving beats, electric guitars, or other noisy-noise, to drive everyone else in the room kookamunga. I would let T1 and T2 play them, if I could get my fingers untangled from trying them myself :)
Thought you might like to know . . .