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I've tried several (e.g., Echofon Pro, Tweetdeck, TwitBird Pro), but I am keep going back to the official Twitter client. Although it has poor list support (and settings UI is strange), it's fast, attractive, priced right (free), and has IMO best-in-class push notification.
 
I still love Twittelator Pro. It's UI is a bit dated but there is not one client with more features. SimplyTweet is my next favorite. I look for 2 things in an iPhone Twitter client: I like to be able to read tweets, conversations, links and pics without having to go deep into menus. Twittelator allows this with one click links, pics, conversations, reply, reply all, etc from the timeline. Next I look for a great compose screen. Twittelator has everything again 1 click away. Insert hashtags, pics, voicenotes, emoji, geotagging, etc.

The only thing Twittelator lacks is push but Boxcar suffices nicely. Also the overall UI could use a refresh to be more modern looking like some of the other clients out there. One thing I didn't mention is themes. If you like themes there is no better client.
 
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