I've heard that phone cameras can't take good pictures of the night sky because the hardware doesn't allow the changing of exposure times. Are there any phones that do allow good exposure for decent night pictures?
Exposure time is less the problem for low light photography than sensor size. Small sensors use small lenses and don't collect a whole lot of light, thus they require exposing for far longer even if long shutter times are supported. This results in excess noise build up and if noise accumulates faster than signal, then the only option to try and get around it is to do averaging and hope the noise cancels itself out enough that the actual signal emerges from the mess. (Image stacking.)
Fundamentally though, the problem comes down to sensor size and noise much more than it comes down to exposure control.
Often, the default camera apps on camera phones won't allow you to make your own exposure choices, but there are many apps out there that will allow you much more control. 645 Pro on the iPhone for example. Search your app store or google some reviews of photography apps to see which ones can help you achieve what you need.
If it's specifically long exposure you're after (for the night sky) there's an iOS app called Slow Shutter that may do what you want.
There are some phones that allow you to control white balance, exposure, etc. and can even output DNG raw files for later processing.
Red Bull recently partnered with Nokia and ran a competition to push the limits of photography using their Lumia range of smartphones