'Play' and 'play with'

Is there any difference between 'play with' and 'play'? Can we say 'play a doll 'or 'play with a doll'?

We play a game, sport, music or musical instrument.

e.g. I play Scrabble with my parents.

e.g. Can you play golf?

e.g. This band played some of our favourite songs.

e.g. I can't play the piano. (Remember to add 'the')

We play with toys and people. 

e.g. The baby played with her teddy bear all day. 

e.g. I want my son to play with other children more. 

e.g. Do you want to play with me?

So, you can't really 'play a doll', unless it became a musical instrument!