You can use the special ability of Babylon (side B, 2nd stage) to get a 7th turn where you are allowed to play your last (7th) card (which would otherwise be discarded). You can use this ability in each age (if the 2nd step is built).
How does this work in a 2-player game where the 'Free City' (NPC) uses Babylon?
Let's say the 2nd wonder stage is already built. It's the 6th turn. Player Bob manages the 'Free City'. After playing his and the NPC's card, he has left 1 card on his hand. The 'Free City' pile has 1 hidden card left. Player Alice has also 1 card left on her hand. Now the NPC gets a 7th turn.
- Bob plays his last card
- Bob draws the last pile card and chooses which of his two cards to play
- Bob discards his card, draws the last pile card and plays it
- Bob transfers his card to Alice, and Alice …
- discards her card and plays the one handed over by Bob
- discards her card, draws the last pile card and chooses which of her two cards to play
- Alice plays her last card
- Alice draws the last pile card and chooses which of her two cards to play?
- Alice discards her card, draws the last pile card and plays it?
(did I miss a possibility?)
Note that even if there would be no choice which card to play, it's still important which player has to play it for the 'Free City' (because this player has to decide whether to build the card or the 3rd wonder stage).
Also, if the last card of one of the players has to be played, it's important which player's hand is used, because one of the players might have the wonder Halicarnassus (which allows to build discarded cards).