I agree with everyone else here. Tweet Bot is the most polished twitter app out there IMO. Nice animations, nice colors, great overall design and usability.
Yeah. You have to log in to twitter in the phone's settings, but then you're good to go. You can post tweets from your app or you can just tweet photos from the photo app without using the app at all. They both work.
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When I turn on notifications in the twitter app they come through as text messages. Is there a third party app that allows you to subscribe to a users tweets through push notifications only?