Main character Joe gets transported from the modern age into medieval times. In Joe's pocket is his smartphone and a solar charger to keep it powered indefinitely. The internet and GPS are gone and unfortunately Joe didn't download a local copy of Wikipedia, maps, or anything unusually prepper/survivalist like that. What common smartphone applications would people or governments find the most valuable? Photos? Video? Calculator? Music? Flashlight? Addictive casual games? Fart sounds? What medieval technology or process would be most enhanced with a smartphone's basic capabilities?
Your Joe arrives in a middle age location, where the normal approach toward stranger is usually suspicious. Then your Joe takes out of his pocket (willingly or upon inspection) a black flat stone, which seems to be able to
- generate light with nothing burning in it
- produce sounds or even music with no musical instruments nor musicians in it
- reproduce images of the surrounding with no painter in it
- talk and answer question when asked (depending on the model) with nobody in it
when Joe does something on it with his fingers. This is clearly a devilish device and its bearer, Joe, is worth a quick trip to the closest Inquisition tribunal to be burnt to death or even lynched on spot.
Based on this premise, the most useful functions of the smartphone are
- lock screen
You'll need to be really careful before you get hung as a demon/wizard wielding a magical weapon...
- having a camera, and being able to record and take videos/photos is invaluable as a tool to gather intel/entertainment
- Flash light
- as an accurate time keeping device (clock, timer)
- you can have a "bard in a hand" with the music player (even a simple ringtone would do)
- for taking and jotting down notes (no need for ink)
- having a chess game installed, with AI difficulty set to impossible will trigger a bunch of chess grandmasters of the time
- and there's A LOT more you can do, depending on the apps you have installed
But the main problem still remains: just don't get killed for being a heretic/demon/wizard/other-worldly being, or get the lords of the land searching for you and your mobile phone.
I advise against selling/trading your phone away, it would make "joe" a regular peasant.
TL;DR; Basically any function a phone has in the modern world, minus the GPS/internet, times 10.
The main thing that Joe would find useful would be buzzfeed and it's "100 things that would kill you in medieval times because you are not vaccinated against them. The 101 is stench".
But if you rule this out what would be most useful is the calculator. And it's preinstalled.
Just look at your screens and see how many mathematic things were discovered after the medieval times (even the late ones).
Also - JOE HAS A FRIGGIN LOGARITHMICS MACHINE! Count the ship out of everything. Artillery angles, powder power, wall thickness, pumps litres per minute, cubic metres of materials needed for anything.
Instead of guessing and making trials by errors - count. Count and give proper measurements and force push those motherlovers into industrial produced things.