Instagram adds fun photo reply feature to its DM interface
Still, it's a bit odd that a service, which is mainly focused on picture sharing doesn't do more with pictures within chat. The latest update, however, might improve on this.
Note that you can only respond to picture messages that are “permanent” (the user has imported them from the camera roll). If you use the camera icon in the Instagram DM inteface, you are sending self-deleting messages — you can not reply to those. A photo reply is also a disappearing message, so you can't create a thread of infinite replies.
Things that are NOT allowed: