Continuing the discussion from Welcome! Introduce yourself here!:
Hi all!
I had thought about opening a post about living without Google Apps, so @jerry1970 set the ball rolling.
First of all: I use only one App-Store at the moment, namely F-Droid (For all F-Droid apps, which work for FP1(U)/2, go here). Besides I have Blankstore installed, but haven’t used it yet. Some apps I downloaded from Evozi APK Downloader or from their respective webpages.
F-Droid features only 100% FOSS (Free and Open Source Software), but it also reminds the user if an app advertises non-free add-ons (like Fennec F-Droid / Firefox Mobile).
Blankstore lets you access the Play Store without installing Google Apps. I have not tested it.
With Evozi APK Downloader you can paste a Google Play Store link and download the apk-file independently from an app store.
So here are the most of the apps I use, with a description and where to find them:
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/system/app mover (F-Droid): This app can make normal user apps to system apps (e.g. if F-Droid is a system app, you can uncheck “Unknown Sources” in Settings > Security.). It needs Busybox to run.
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Add to calendar (F-Droid): Used to import individual ics-calendar-files (like from the regional transport webpage anachb.at). For online iCalendar files I use ICalSync2 (see below).
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AFWall+ (F-Droid): A simple Firewall, which I use to block internet access from various apps, among them the stock browser, which is vulnerable because of the Webview bug.
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Amaze (F-Droid): A beautiful file manager with root privileges. With it I can access all system files, open zip-files and modify text files.
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Autostarts (F-Droid): I use it to block some apps from starting automatically, like the default E-Mail app (I use K-9 Mail - see below).
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Barcode Scanner (F-Droid): Good app for reading QR-Codes (also reads calendar-QRs or contact-QRs).
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CatLog (F-Droid): Read out system logs, search for keywords and create text files of logs.
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DashClock Widget (F-Droid): It is a nice way to display notifications and the weather on the lockscreen.
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DAVdroid (F-Droid): I used it to synchronize my Fairphone 1 to my owncloud caldav server and subsequently to Thunderbird on my Windows 7 desktop.
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Episodes (F-Droid): @paulakreuzer made me aware of it here. Episodes lets you keep track of your favourite TV shows (Currently I am watching StarTrek: TOS, Season 3, from 1969
).
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Fennec F-Droid (F-Droid): I think it is much easier to use than the default browser on the FP1 because it has a nice GUI. Plus it features an easy-read-mode and a read-later-list, where you can save pages offline. Besides you can expand the functionality as known from the desktop version, and there is also a Firefox Marketplace, where you can even download (almost) stand-alone HTML5-apps. (There are some nice free games on there). Fennec F-Droid is not affected by the Webview bug.
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GravityBox (Xposed Module Repository): Let’s you personalize everything! I have an orange notification bar, for example… Also it provides a lot of useful modification to the OS, which make the use of the phone easier and faster. Try it out! (You need the Xposed Framework to run it.)
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ICSdroid (F-Droid): It’s provided by the same Austrian developers, who make DAVdroid. Only this one helps to synchronize *.ics online calendars.
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K-9 Mail (F-Droid): Great e-mail client, which fully replaced the stock e-mail app for me. (I also use K-9 Mail DashClock add-on).
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KISS launcher (F-Droid): A great, simple launcher: Type to find.
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List my apps (F-Droid): Nomen est omen.
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MemoPad (F-Droid): Doodle on your touch screen and share the pictures.
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Mozilla UnifiesNlp Backend (F-Droid): Lets you use Mozilla’s location provider on your FP1(U). µg Unified Nlp is needed (see below).
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MyExpenses (F-Droid): Helps me keep track of my expenses. There is a backup function and a export to excel. Plus a lot of functions, I think you need a business diploma to understand…
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Notable (Evozi): This is a light-weight app to handle your tasks because it displays them in the notification bar and every time you slide down, you get reminded. In the latest version the developer even added alarms, still it doesn’t need any permissions.
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OsmAnd~ (F-Droid): This one is also GREAT: It is a must have because without Google Play there is also no Google Maps. But who needs it, if you have OsmAnd~!? It is a bit more complex to use, but it also has a huge functionality, featuring all the usual functions we know from Google Maps, like navigation. But there is also the possibility to choose among a lot of different maps (like MS Bing-Maps, OSM (OpenStreetMaps), and also country-specific apps). Also you can easily record GPS-tracks and save them as GPX-Files. And since OSM comes with a lot of POIs (Points of Interest), you can display them all on the map, or filter them to display only those you want to see (like I did with the citybike stations). Best feature: You can download OSM-vector maps and store them offline on your phone!!!
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Polite Droid (F-Droid): Great app, which automatically silences your phone during calendar events.
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SatStat (F-Droid): I use it to verify the visible GPS satellites and to check the phone’s sensors.
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Share to clipboard (F-Droid): Doesn’t have a GUI, but adds the option of copying to the clipboard to the share menu (some apps don’t let you share to SMS, but using this app you can at least paste the info manually).
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Terminal Emulator (F-Droid): A command line interface for your phone.
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Titanium Backup (F-Droid): Titanium Backup makes me perform updates more light-hearted because I can be sure that my app data is safely stored. I backup to my SD card, so even if the internal storage gets wiped, I don’t have to set up every part of my phone anew.
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Twidere (F-Droid): A Twitter client (find me through @StefaBrand ;)). Much more functionality than the official one, a lot of personalization-options. I also tweet using the hashtag #austrianfairphoners.
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VLC (beta) (F-Droid): Instead of the default music app, I use this one because I think it displays the music library better. Also you can easily play network streams. And for me it just looks nice :).
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WhatsApp (downloadable through their website): Just because everyone has it…
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WPS Office (formerly Kingsoft Office) (Evozi): Also indispensable for me because it can display all MS Office files and PDFs in one app.
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Xposed Installer (their website): Is needed for GravityBox (see above).
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µg UnifiedNlp (F-Droid): Is needed for Mozilla UnifiedNlp Backend and other backends to provide location lookup, even without Google Services installed.
What are your alternatives to Google Play? Which alternative apps do you most like?
I am looking forward to hear some new ideas!
Greetings from Austria
Stefan
PS.: Maybe we could create a page here on the forum where we collect the must-haves (like in my opinion OsmAnd~)!? Edit3: Here it is! (Kudos to @paulakreuzer) 
Edit: I’m not using 1Mobile market anymore, but kept WPS Office.
Edit2: @paulakreuzer provides a very complete list of his (mostly open source / free) apps here.
Edit3 on 28.08.2015: Complete update.
