One activity that gets everyone involved is a dance party. Make a CD of movement songs such as "YMCA," "The Hokie Pokie" and "The Electric Slide." Then, learn the movements that correspond to the music and teach it to the participants. Everyone can get involved at once and as soon as everyone has the moves down, you can make the activity into a game: a dance off. Rotate who dances at certain parts of each song.
Simon Says is another game that can involve everyone at once. It consists of a leader, "Simon," who instructs the rest of the participants on what to do. The participants have to follow directions or else they're removed from the game. When dealing with mentally handicapped adults, start with simple everyday movements, such as walking, stretching and jumping. You can even play the game to a catchy song to help participants follow along better.
Scavenger hunts can be done either indoors or outdoors, depending on the development level of the group of mentally handicapped adults you're working with. Place the participants into small groups and hand each group a piece of paper consisting of things that they have to find either in the room or outside. These can be things that you place around the area for the groups to find or things in their natural environment. The first group to find everything on the list wins the game.
Musical chairs is a game for all ages, from children to adults, mentally handicapped or not. Arrange chairs for each game participant and have the players circle the chairs to music. When the music stops, that's the sign for each participant to find a chair. Those who don't are removed from the game. As the game goes on, the game facilitator continually removes chairs until there are two players left vying for one chair. The one who sits in the last chair available is the winner.