There’s a number of physical challenges that can be suggested, at the risk of seeming ableist, such as pinkies only, thumbs only, sitting down, standing on a platform, cross armed (aka lotus), standing with your back to the table, wearing an eye patch.
Per-game objectives can be easy to suggest and monitor, although they may vary from game to game. Many have been mentioned above. Some are time-based, objective-based, or a combination of both.
Finally, consider “hot potato”-style rotation games, which are not necessarily limited to 4 players, such as Stall Ball (see Stall Ball )