nice topic there Shrike, thumbs up for a fun idea.
Ios games (and perhaps Android ones too), are a rather broad category to pick five from, being as there is everything from complex rpgs to short arcade games to puzzles, but I'll have a go, again no particular order here and no promises as to any sort of inherent logic.
1: King of dragon pass (I bet nobody is surprised). Part gamebook, part strategy, part rpg, part history, all fun! difficult to get into as Shrike said though well worth the effort.
2: Fate haven. There are of course god knows how many choiceofgames and hosted games these days kicking around, probably over a hundred. I haven't played all of them by any stretch of the imagination, but of those I've played Fate haven really stands out. There are games with more choice (choice of robots), there are games with more battles and magic, and there are games with longer more complex plotlines, but Fate haven just manages to strike that balance just right.
It also manages to do something is if not impossible, certainly bloody difficult, include probably every fantasy cliché in the book, peasant from tiny farming village (which of course gets destroyed), mysterious stranger, elemental magic, standard D&D type weapons and attributes, not to mention all the expected races, mercenaries, rollocking adventures with evil sorcerers and insane religious cults.
yet, for some reason it gets everything just that bit right, from it's characters to it's relationships to the final understanding of the world and even a slightly fourth wall breaking but amazingly appropriate ending.
3: A dark room. I had to toss up between this and the ensign, and indeed the adventure to fate games. Dark room won for it's absolutely and story telling and the fact that the rpg mechanics are just so dam logical and simple. I'll not say anymore, but if you love rpgs, with addictive worlds, grim writing and a uniqute flavour you will love this, it's sequel, the ensign is good too.
4: Blindfold breakout. Play breakout with your phone as a paddle. Arcade action at finest, and a truly unique experience.
There are a lot of blindfold games (most still need entries on this site), and while I don't always agree with some of the design decisions, the actual quality of the work done is definitely there, and this especially shows in the arcade games, this is the one I tend to pick up for a quick blast, though I confess when picking my favourite I had to work hard to decide between this and blindfold hopper.
5: Pappasangre ii: If indeed this has been abandoned by it's developers that is bloody sad! the culmination of lots of work with awesome audio atmosphere and great control, definitely a real classic, especially given the narration by sean bean.
Football chairman is not a game I know, me not being a sports person, though I'd be interested if you wanted to write a description for new releases.
Sixth sense I liked, but it was rather beaten by audio defense, which managed to do a similar thing but with unlockables and 3d movement.
I love crafting kingdom, but the slowness of navigating the interface is putting me off the game rather, when the bugs are fixed I'll definitely get back to the game.
other games I like but didnt' mention include star traders elite, brilliant game, but I do wish they'd fix the planets list so you can know what is where around you and what belongs to what faction, the amazing adventure to fate rpg games, which are wonderful to play but definitely complex in their combat, tactics and equipment, the amazing time crest which I deeply enjoyed up to the point I seem to reach an unwinnable situation in the fourth chapter, the very good turn based card game lost cities, and from the choiceofgames/hosted games list, mecha ace, samurai of Hyuga, and choice of robots.
With our dreaming and singing, Ceaseless and sorrowless we! The glory about us clinging Of the glorious futures we see,
Our souls with high music ringing; O men! It must ever be
That we dwell in our dreaming and singing, A little apart from ye. (Arthur O'Shaughnessy 1873.)