A list of animals that can't play league
Dogs: https://media.giphy.com/media/mEtfDhbWwc4GA/giphy.gif Dogs have no thumbs, and thus can't pull of any LCS big plays. They can press R though, and thus are quite proficient with Karthus.
Goldfish: http://cdn2.hubspot.net/hub/1759064/hubfs/Blog_Images/Gif_Reactions_Optomised/goldfish.gif?t=1458044961544&width=514 Goldfish are fresh water fish, and thus cannot handle the salt of league.
Cobra: http://i.perezhilton.com/wp-content/uploads/2014/08/spitting-cobra-edit.gif One would think that, being Toxic, that cobras would fit right in with the league community. But alas, they always try to eat their mouse, and thus can't control.
Boosted Animals: http://www2.b3ta.com/fp-archive/host/9932059-1.gif 'Nuff said.
Are there any other animals that can't play league? Comment below!