Much of what I do uses Apple’s own apps, and I can’t think of many, if any, cases where I have replaced a default app with something else. I’ve limited this to iOS, in line with Shaun’s post.
Mail: it would be better of it had PGP support, but it’s good enough, especially now that it supports multiple signatures
Safari: it works fine. I use an “always-on” VPN when not connected to a trusted Wi-Fi network, so all traffic routes back through my servers, so I get the benefit of network-level tracker blocking, access to services on the home network.
Reminders, Notes, Camera, Podcasts, iBooks etc. All fine for my needs.
Other apps which I use frequently are (in alphabetical order):
1Password (iPhone / iPad): I have no idea what my passwords are, thanks to 1Password. Essential on both iOS and macOS, in my opinion
Authenticator (iPhone): a good two-factor authentication app, although it doesn’t seem to offer backup, which is a shame
Blink shell (iPhone / iPad): excellent ssh application
Glympse (iPhone): great for sharing location with friends who don’t use iOS
Good Notes (iPad): for taking notes with the Pencil, with different notebooks per topic
Groundwire (iPhone): the best SIP client available for iOS, in my opinion
iAnnotate (iPad): sadly, WebDAV support was removed from the new version, so I am stuck on version 3, but it is still my go-to PDF reader/annotater
Nextcloud (iPhone / iPad): client for our Nextcloud server. Think “Dropbox, but not on someone else’s computer”
TomTom (iPhone): their new app/model is a good example of how to take something great and ruin it pretty much comprehensively. But I’d still rather have this than rely on something which needs a cellular connection. Note: the original version is available here.
TouchBlur (iPhone): for blurring out faces and so on in photographs, usually for when I want to tweet something but remove unwanted bits
Tube Exits (iPhone): because I still don’t know the best routes to use on the London Underground
TweetBot (iPhone / iPad): Twitter client. It would be better if it could do scheduled tweets