I love hearing about unique takes on game mechanics. Someone recently convinced me that limited inventories are kind of abused currently and that unlimited inventory systems would give more player choices.
I love simple controls or an elegant way to control simply. For example using one thumb to control two buttons simultaneously or the Super Mario Run control scheme where you only press on the touch screen, doesn't matter where, and that's it.
I hate it when in co-op game the other player's actions can screw up the game e.g. moving the screen too far so the other player dies.