Help!

Hello Astonishing, and welcome to the forums.

For the future, it's better to give your post a descriptive title and not just write "Help!"

I would use "play." It's parallel to "do."

Is he doing anything besides playing games?
Does he do anything besides play games?

Thank you

Is this a common mistake in informal speech?

I wouldn't even call it a mistake. In speech, people change direction about where their sentence is going all the time. If you ever read actual written transcripts of what was said, you'll find it highly "ungrammatical" but perfectly understandable.

Ok makes sense, thanks for your help.