Some ideas of things my kids and I enjoyed as babies off the top of my head (not sure whether you want games specifically or just activities and things)...

Fine motor skills: Ribbons threaded through two sides of an object then knotted to be pulled
Transferring objects from one thing to another (pasta, rice, lentils, pompoms, buttons etc). Start with hands then introduce other things like spoons or tongs.
Hammering golf tees into styrofoam
Pegs
Threading pipecleaners in strainer
Posting paper into slots or blocks into holes or coins or rocks (shape sorting)
Stacking - cups, boxes, blocks etc
Tearing paper

Sensory: Jelly with hands
Cooked spaghetti (with or without paint)
Finger painting
Sand and mud
Bubbles
Ziplock bags filled with different liquids (shaving cream, liquid soap, glitter water etc)
Texture cards

Music: Shakers with rice, rocks, bells etc
Pots and pans for drumming
Rubber bands taut over boxes for plucking

Miscilaneous: Hiding objects underneath things, peekaboo
Mirrors
Photos of family members
Hiding objects in bottles of rice so they need to twist and turn it around to find them (more advanced is magnetic objects and using a magnet to find them)

I recently made DD2 a busy board which has been popular and could be adjusted to suit younger (or older) ages. Here it is and in action.