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My favorite is one that seem to be unknown to most...

https://itunes.apple.com/us/app/backgammon-online-2/id395137246?mt=8

It also allows you to play online at FIBS (first internet backgammon server-play bots or real people)

I have been playing at FIBS for about 15 years and have over 5,000 matches played. I used to use a Mac app, first classic then OS X, but now play almost exclusively on my iPad or 6 Plus with the app above.

For whatever reason this app never seems to show if you do a search for backgammon in the App Store on your ios device, not sure why.
 
It is pretty good, but I think backgammon masters takes the cake.

The app has had a few good improvements for the last year or so, the internet game seems faster, too.

I mostly just play hotseat at evenings with my son, but have tried online a few times and it seems solid.

Web: http://mastersofbackgammon.net/
 
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