Try opening the forum from your phone!
Discourse has an excellent mobile interface, and the necessary metadata to be pinned as a webapp on your home screen.
A native app is… probably not happening, since it’d basically be a wrapper around the website anyways. We could do it (or someone could make a generic Discourse client), but that would either be a wrapper, or pretty much just cause you to miss out on our customizations (like our theme switches, and the non-core spoiler tags), for very little actual gain…
(Push notifications would be awesome, but unfortunately a pain to implement with the way both Apple and Google’s implementations work… I’m gonna take a look at Pushbullet though, I may be able to make a Discourse plugin for that or something.)