Ever since buying two android devices in relatively short term, I've been gradually scouring, installing and testing some (mostly free) apps. Here's my 'collection', thus far:
Tools:
Adfree android*: by itself enough reason for rooting.
Alarm clock extreme: it may be a simple alarm clock...but that option to gradually increase the alarm is interesting. Probably not as good as my wake-up light alarm clock, but at the very least miles better than the stock app.
Amazon app store: I'm kind of torn on whether to add it, as it's direct competition for the google play store, and way too active for my taste when I'm not using it. On the other hand: I'm a sucker for those daily deals (for now).
Barcode scanner (from Zxing team): may come in handy one day. Look for these makers: I've tried a few who hadn't had as much support for different kinds of barcodes.
Dolphin browser: I'm sort of undecided on web browsing on mobile devices. This feels speedier than chrome, which is easier to use in other areas. Haven't tested opera yet.
Droidcam: the (cheapass) webcam I had refused to work under windows 8. With this and a litte bit of setup, my smartphone acts exactly the same way (except with way better quality)
ES File explorer: great explorer
Nova launcher: haven't tried that much launchers, but this is a good one for my taste.
Maps.me: I don't have a data abonnement. This app lets me download all the maps from any country, ready to consult offline.
mobicalc free: a calculator. I tried realcalc, but the solution preview in a different window makes it all the better
MX player: my choice of multimedia player
Perfect viewer: a comic + PDF viewer. I choose this for being one app over two, but not sure whether specialised apps have better options
Power toggles: this is a widget that lets you predefine buttons. A 'power control' clone, but with more versatility.
Quickpic: my choice of image viewer. Better than what I had as stock, but haven't thoroughly compared it versus other freeware
Rocket player: haven't tried all the different music players, but this is certainly a solid one.
Swiftkey: both this and swype are pretty decent. Swype even has an edge, except for one important thing: no enter key on text messages (oh, and limited smiley support). But both are miles better than the standard keyboard.
Swissarmyknife: before Power toggles (which has a flashlight as well), this app was worth it for the flashlight alone. But it has 9 other nifty measuring apps (eg. a compass, leveler, mirror and magnifying glass...even the calculator isn't bad at all)
Titanium backup:* for obvious purposes.
Ultimate dynamic navbar*: replaces the softkeys with ones you choose. Things like hiding also works. (note: gotta say that this doesn't quite work when it comes to replacing the standard ones, but since all similar apps do this, it's something in my android setup)
Whatsapp: only one vote? C'mon...this thing practically DESTROYED payed messages'es around here.
x-plore: another great explorer with dual browsing windows. For this reason, I'd advise it on tablet only
Youtube: why do I even mention this? It comes on all devices nowadays.
*requires root access for full potential
Games:
7x7: a block puzzle game. somewhat a mix between four-in-a-row and tetris, though it isn't time based. Simple rlax game
Badland: flappy birds meets limbo, I guess.
Cut the rope: the classic
Flow free: draw lines so the entire board gets covered without any cross-overs
Flow free: bridges: as previous, but with one or two cross-overs
Fruit ninja: simple mindless fruit slashing. One of the first games that made me realise video games can be fun WITHOUT physical buttons
Glow hockey 2: air hockey!
Also fun to play against a friend.
Linky dots: each island is connected horizontally and/or vertically with 0, 1 or 2 bridges (who don't cross each other). Find out how. A puzzle game.
Move: swipe in a direction to move all balls at the same time. Get them all on their destination in X moves.
Piano master free: guitar hero but on a piano. I wouldn't advice it on a smartphone (or on anything below 10", for that matter)
Piano tiles (don't tap the white tile): far less piano, far more a reflex game. But a fun and addicting one.
Power lines: connect all the houses with power by rotating the pieces on the grid. To pass the time
Plants vs zombies: way too addicting. 100%'ed this on steam...now working on the mobile device
Smash hit: an on-rails...shooter kind of game
Sudoku Free or sudoku 2go: the latter has the most options, but the rather minimalistic design (and google play integration) makes me play the former the most
Temple run 2: like Temple run 1, I guess
Threes: like 2048, but with stacking threes. Nice theme.
Sidenotes:
Kingo root: a windows program that'll make rooting your device painless (assuming it's supported).
(final note: titanium backup, plants vs zombies and threes are the only paying apps. The others sometimes have restrictions, but none really bothersome or hindering)