An AI that asks questions? Typical AI these days are question-answering machines. For example, Siri, Alexa and Google Home. But it is always the human asking the questions and the AI answering. 

Are there any good examples of an AI that is curious and asks questions of its own accord?
You are referring to 'proactive AI' as opposed to 'reactive AI' like Alexa, Cortana, Siri, Bixby, Google Assistant, and others.  There hasn't been much progress in this area of AI.  Google's recent demonstration of Duplex addresses this to some extent.  Some chatbots are proactive. Genesys provides such capability.  Check out their video

Azure's bot service has a page on how to implement proactivity and there is another video that walks through the whole process: Learn to build Proactive Bot in 30 Minutes.