Kingdomino - excellent excellent domino type game (but much more to it than dominoes). Very addictive and good for 2 players.
Jaipur card game as per pp above (which is also loved by my 5 and 8 year old)
Patchwork - really addictive game with a small box where you compete for Tetris type pieces and have to fit them together on a grid.
Akrotiri - really attractive game where you place map tiles and (a) try to make patterns in accordance with certain objectives and (b) create shipping lanes for your little boat to move around and trade.
Agricola (DS2 8 plays this too) All Creatures Big and Small, though 2 people can play the big version very satisfactorily Backgammon (both DSs play) Bezique Cribbage Caverna (v heavy if sending abroad though) Settlers of Catan the card game (DS2 loves this too, and Jaipur) Isle of Skye (2-5 players, unusually).
Jaipur - beautiful aesthetics, easy to grasp, fun and addictive (a bit fiddly to set up though). My DC have enjoyed this since they were 8 or so. Battle Line - this is our favourite. Simple rules and almost unbearable tension. Port Royal plays very well 2-5, our DC (9 and 12) love this as well.
We used to play Monopoly Deal as both 2-, 3- and 4-player. Brilliant little simple and addictive card game that we must have played hundreds of times.
As for Lost Cities - I love this but DH and the DC find it too dry.
We had Agricola All Creatures Big and Small but sold it. Deeper and more static/strategic than the others recommended above which are more simple and more dynamic/tactical.
We have Patchwork but didn't like the tetris/spatial aspect. Lots of people love it though, so worth considering whether the spatial aspect would appeal or not.
I also love Carcassonne as a tactical 2-player game, but can't get anyone else in the family to play.
This year I have bought Hanamikoji for DH, it is supposedly a brilliant two player game. Possibly a bit deeper than Jaipur, Battle Line and Lost Cities.