Around age 2,5 years we started to play boardgames with our oldest son. Not playing with boardgame, it's details but really playing and respecting rules. The first board game of Jacob became
Ballons. It is a card game suitable also for children aged 3, publisher -
Amigo.
The game can be played by 2-5 people, and this is quite short (on the box written: ~ 10 minutes). Game mechanics is quite simple: each participant has 5 cards, they are placed on table face up.
Game action consists of drawing action cards. Possible actions are (1) loosing balloon (card is put face down) (2) gaining new card or opening face up the old card.
The winner is player with most balloons.
Playing the game with child, kid repeats and learns basic colours and counting till 5 or 6. It's also good way to teach child that you will sometimes something loose in life and you need to get along with it.
And, of course, game is a good opportunity for child to deal with loosing something, in this case, for example, cat snaps balloon or balloon flies away in the air.
When we had taught the game, the child asked to play it every night, but eventually dropped the child's initiative and now quite rare he pulls it out.
Comment to real boardgamers: in principle this is a simple luck game, because the ability to influence a game is minimal.