Here's an update to the version of this list I wrote at the end of 2013. This is from my Android-centric view of the world, and leaving out the obvious ones like Gmail, Play, Maps and Calendar—though I will note I ditched Gmail for Google Inbox, for personal mail at least. A lot has stayed the same, though social networks have come and gone, and the wearables world has advanced.
Slack — now an integral part of my work and social lives
Swarm — replaced Foursquare
Facebook — (new!) I reversed my departure from Facebook, mostly to keep in touch with family
Instagram — show me what you cook, I really care!
Twitter — surprisingly, holds up well. Mostly professional use these days
WhatsApp — to stay in touch with European family and friends
No longer used: Path and Google+ (alas)
Evernote — my outboard brain, and importantly a universal inbox
LastPass — lifesaver, repeatedly, but...
Dashlane — now preferred to LastPass for password management
Todoist — a fan over many years, however much I look around, I keep coming back
Worldmate — my preferred travel companion app, replaced TripIt
RescueTime — slaps my wrists every time I spend too much time on Twitter
Glympse — a bit of a reach for this category, but great for contending with the vagaries of commuting in the bay area
No longer used: Any.do (can't leave Todoist), TripIt (too expensive for what it delivered)
Hue — still a terrible user experience, but we buy more and more bulbs!
Android Wear — my Moto 360 watch replaced all the fitness bands
Wiithings — keeping myself honest with the scales, nice incremental improvements this year
Chromecast — we now have three. Best AV purchase ever
Netatmo — received a well-overdue UI overhaul
No longer used: Jawbone Up
Strava — the best app for tracking running
RunKeeper — I keep this in sync with Strava and use it to track other workouts
LoseIt — one of the earliest fitness apps on Android and still working great
Google Fit — slowly becoming more useful at integrating fitness information
Sleep As Android — works well with Android Wear as alarm clock and sleep tracker
Amazon — still the place I buy most things online
Netflix — we love our Chromecast
Google Play Music — complete convert to All Access for a while now
Plex — where I digitize my DVDs to
Finally, a word about iOS, which is the same as I said two years ago. I still have and use an iPad, but for one thing only: FiftyThree's most excellent Paper app. I added a Pencil too.