I can give you a summary of my own and other Tonmo members' experiences.
Octopuses have different personalities, so their way of playing can vary, too. Also, they become less afriad as they get older and may interact more and let you pet them.
Personally, I think octos resemble dogs in what they like to play. Tug of war with the feeding stick is a favorite (and you don't stand a chance of winning!).
Steal and hide the cleaning magnet is another game. Then your octo may dangle it out on one arm so that when you try to get it back, your octo zaps it back into its den.
Another method of play is playing by itself with objects that you provide, like Carol's Ink with the lego block.
Also, they may chase your fingers as you move them on the outside of the glass.
Maybe our other Tonmo members can volunteer additional ways that octos play.
Nancy