A game for an almost-2-year-old

I am looking for a board game type of game:

1. something that is fun (duh!)
2. that is “conceptual” in that it requires concept-formation, forward-thinking, memorization, etc.
3. that can be played in a group
4. that is appropriate for a clever and patient 22-month-old.

The social aspect is very important: I feel she needs and would welcome something interactive with other people  (so none of those “I’ll read to you” or “I’ll play with you” machines), and even children (she is still a very parallel player with kids her own age. For closer interaction needs the kind of directed attention that only adult and older kids can give her.)

We engage in a lot of play together: we diaper her bears, “clean” the house together, build towers with blocks and Wedgits, etc. But I am looking for something less physical, something that will bring us together in a more cerebral kind of space. 

I love to see concepts “light up” in her – like they were already there, in her brain, and they just needed to be switched on. This morning, for instance, I asked her: “What is the difference between Mama’s arm and Baba’s arm?” (which she likes to pinch when searching for that ever-elusive sleep). She thought for a couple of seconds and said: “Hairy”. So she understands the concept of “difference”.

This game should allow us to discover and exercise such cognitive skills like matching, spotting differences, concentration and memorization.

We do that when we read stories together, when we go through “spot the balloon” kinds of books. But we now need a game in that it should allow her to manipulate the events, move things around, which will give her sense of decision, of realization of her own change-making capabilities.

As such, it should also make her aware of the consequences and responsibilities of that kind of power, and make her more foreward thinking, more calculative, with plans of action, etc.

You know what I mean, right?

Board games, of course, is what comes to mind first, but most of them are beyond her as yet. There is one that sounds promising, though: the Goodnight Moon Game. Has anyone tried it?

Or am I asking too much? Should I just design our own boardgame?