Rüssel raus!, for example, is a game by Inka Brand, Markus Brand, and Matthias Prinz for 2-4 players, aged 6 and up, that has everyone involved on every turn:
• Da bockt der Bär (The Bear Bucked) from Treo Game Designers features a similar level of player interaction for 2-5 players, aged 5 and up, with you helping to determine whether your neighbor moves like a mouse, goes like a goat, or barrels along like a bear. In more detail:
• Players mess with one another a different way in Einer geht noch! (Cruise or Lose), a Paco Yanez design for 2-5 players, aged 8 and up, with you trying to sink opponents, while also working with them when you end up in the same boat. An overview:
You draw three animal cards per round. You play these into boats: twice face up and one time face down. Each boat holds only three animals, though. After all of them have boarded, some animals can cause others to switch boats. Any boat with passengers that, in total, are too heavy for it will sink. Animals aboard boats that don't sink are worth VPs. If you have the most VPs after four rounds, you win.
• Finally, we have a game in the "do things quickly at the same time as everyone else" bucket: Flotter Otter, a Daan Kreek design for 2-4 players, aged 8 and up:
Each round you reveal one new card for all players. You can order the objects shown on each card in two different ways, e.g. clocks might have varying sizes and display different times of the day. Since your bricks correspond with the unique colors of the objects on the card you can show one of the two possible solutions by stacking your bricks. The faster you are, the better — so long as you don't make a mistake. This also means that it is smart never to give up early, even in rounds when you seem to be lagging behind.