There are not many apps formally labeled as Orthodox Christian, but there are some apps and mobile websites that can be used in the pursuit of the spiritual discipline of ascesis. Among these apps are:
The value of this app goes more or less without saying, but there is one caveat.
I visited a monastery whose rules included not playing recorded music, and I saw outside the nave an old man, with headphones on, listening to Byzantine chant, moving to its beat and off in his own world.
It struck me, if anything, as an act beneath the dignity of an old man. Being off in your own world is not good for anyone, but there are some things tolerated in youth that are just sad in a mature adult. And as best I can surmise the rule was not a rule against certain types of prohibited music (though acid rock would be a worse violation), but using technology to be off in your own world in the first place. And it is because of this that I rarely listen to recorded liturgical music: it is easily available, but there is something about it that is simply wrong. (And this is not just a matter of how digital music sounds to someone with perfect pitch.)
I know of nothing better in terms of Orthodox Internet radio than Ancient Faith Radio, and really have very little if anything to say how Ancient Faith Radio could better do the job of a radio station (or, as the case may be, two stations providing access to lectures, sacred liturgical chant, and access to past broadcasts). But I have some reservations about why Orthodox need to be doing the job of a radio station—as, for that matter, I have a cautious view of my own website. The ironically titled The Luddite's Guide to Technology lays out the attitude where a radio station with crisply rendered sacred song available on one's iPhone or Android should be used that much.
The Psalter is the greatest prayerbook of the Orthodox Church. This page selects a Psalm at random, until you click a link and it provides another Psalm at random. They seem to be helpful, chiefly because the Psalter is helpful.
I hate to hawk my own wares, but I've put quite a lot into the two sites linked above, and perhaps reading them may be of some use.
JonathansCorner.com/clock/phone.cgi (bookmark for iPhone, iPad, Droid, Samsung, Android, Kindle, and Blackberry, both phone and tablets)
This is a jumping point to your local liturgical day's readings from Scripture, saints' lives, and troparia/kontakia. It is intended to follow your local liturgical day, from sunset to sunset, which is why it asks what city you are nearest to. It offers a choice of Old Calendar or New Calendar, although the Old Calendar has room to improve in terms of dates relative to Pascha. It is a jumping off point to the real OCA trove of readings, saint's lives, and troparia; much more than the lion's share of the value is in the OCA feasts and saints to which this provides a jumping off point.
Evernote (iPhone and iPad users may consider Instapaper) offers the possibility of a library of texts to pray. Of course it can be many other things besides a library of prayers, but you can store morning and evening prayers, preparation for Communion, and a wealth of Akathists. There is much more that they can be used for, but I've found it well worth my time to type up liturgical prayers and have the text available for prayer later.
Your smartphone's built-in note-taking application, or Momento for iPhone or iPad (with many diary applications for Droid, Samsung, and Android, and perhaps other offerings for other devices.
Some monks in the ancient world kept a notebook, and something to write with, by their belts. They would stop at intervals to write down their thoughts: not brilliant ideas to think about, but take moral stock of where they were and how they were doing.
Such a practice was not mandatory in the ancient world and to my knowledge no one requires it now. However, since I started doing it, I have besides some very stupid struggles come to a higher level of awareness, of nipsis, of how much goes on in my head and heart that is simply silly.
If you do this, it would be better to have an application that stores your information locally in your smartphone instead of uploading it to the cloud where others may more easily find it. That is why I don't recommend Evernote for this purpose; it is a very attractive app in many ways, but not as strong on privacy as your smartphone's built-in note taker or a diary app.
The Kindle app for iPhone, iPod Touch, Droid, Samsung, Android, and Blackberry (not to mention PC's)
I wouldn't want to denigrate paperback books, but you can buy some of the greatest classics on Kindle: the Orthodox Study Bible (an edition I discussed in my Orthodox Bookshelf), the Philokalia, and My Orthodox Prayer Book. Plenty of lesser works are available too: see my own Kindle offerings at amazon.com/author/cjshayward.
The icon library at IconLibrary.mobi.
While preparing to write this, there was one thing I looked for but couldn't find: an icon library, or at least an icon library that features Christ, the Theotokos, and the saints and angels instead of just featuring angels. So I created IconLibrary.mobi. It's better than nothing.
G.K. Chesterton said, "There is more simplicity in the man who eats caviar on impulse than the man who eats granola on principle." And there is more Orthodoxy in just using an iPhone as a tool for being human than going to the app store and looking for apps endorsed as religious.
Do a Google search for best iPhone apps, for instance, and you will find a wealth of app recommendations. And many of these can work as a support for an Orthodox life of ascesis: Orthodoxy did not invent the pot, the belt, or the hammer, and yet all of these can have a place. Not, perhaps, the same place as a book of prayers, but a legitimate place. Secular tools and activities are holy when they are used by a life out of ascesis. And there are some excellent apps; among them I would name PocketMoney, a personal finance tool; mSecure, a password manager; Things, a to-do list; GPS MotionX Drive, a navigation tool; Flashlight, which lets you use the LED as a flashlight; and the main Google app, a search app optimized for the web by the company that defined search. All of these have their place.
But there are many apps on those lists that are unhelpful. Games and entertainment apps are meant to kill time, which is to say that they are meant to provide a convenient alternative to the spiritual discipline that tolds monks, "Your cell [room] will teach you everything you need to know." And I suggest asking, in considering an app, "Does this support ascesis?" A to-do list helps with nuts and bolts of a disciplined life. An app to show a stream of new and different restaurants feeds gluttony. It is fine for Orthodox Christians to use apps that are not branded as Orthodox or ascetical, but the question "Does this support ascetical living?" (which is in no way a smartphone-centric question, but a basic question of Orthodox life, to be settled with one's priest or spiritual father perhaps), applies here.
There is a good case to be made that the most important number for you to be able to call on any phone is 9-1-1. However, this does not mean that your first stop in dealing with boredom is to call 9-1-1 to just chat with someone. It may be the most important number for you to be able to call—and the only number that may save your life—but using 9-1-1 rightly means using it rarely.
I would not speak with quite equal force about smartphones, but I would say that if you really want to know how to use your smartphone in a way that supports Orthodox spiritual discipline, the biggest answer is not to use one more app or one more mobile website. It is to use your phone less, to visit people face to face instead of talking and texting, and to use apps a little more like you use 9-1-1: to get a specific task accomplished.
And this is without looking at the problem of an intravenous drip of noise. The iPhone and Android's marketing proposition is to deliver noise as an anaesthetic to boredom. And Orthodox use of the iPhone is not to deliver noise: all of us, with or without iPhones or Android devices, are to cultivate the ascesis of silence, and not make ourselves dependent on noise. And it is all too easy with these smartphones; they are designed so that it is too easy.
The Fathers did not say "You cannot kill time without injuring eternity," but on this point they could have. Killing time is the opposite of the ascesis of being present, of being attentive of the here and now that God has given us, and not the here and now that we wish to be in or the here and now we hope to get to. Contentment and gratitude are for here now, not what we imagine as better conditions. We are well advised to live astemeniously, and that is where the best use of iPhone and Android smartphones and tablets comes from.
Smartphones have many legitimate uses, but don't look to them for, in modern terms, a mood management tool.
The Best Things in Life Are Free
iPhones and Spirituality
The Luddite's Guide to Technology
Technonomicon: Technology, Nature, Ascesis