Maybe a stereographic projection, or Ptolemaic planisphere, would be a fun way to display a local map on your cellphone. You’d still have different possible zoom levels, but all of them would show the whole world on your screen, except for an arbitrarily small area just around the point of projection. You could move the point of projection as you moved around.
Planisphere projection has many interesting properties. It’s conformal, which is to say that it’s locally planar — it doesn’t distort the shapes of small objects — but it’s not isometric or equal-area, which is exactly what would seem to offer the possibility of making a potentially useful local map that includes the entire planet.
There’s the question of how to orient things. You can try to orient the map with north at the top, or you can try to orient it according to the cellphone’s compass, but either way you have a contradiction. Suppose you orient it with north at the top. In the center of the screen you have the antipodes, and around it the rest of the world, and you are positioned at the point at infinity. Suppose you’re in a cul-de-sac with a street leading out to the north. Does this street appear at the top of the screen or the bottom?
If it appears at the top of the screen, the more northerly points on the street are further down the street. The least northerly point, the end of the street where you are, is at infinity. More and more northerly points are further down the screen toward the antipodes, until finally the street ends. So in fact north is down.
Consider, though, a parallel street far enough to the west that its end appears on your map, which also ends in a cul-de-sac just to the west of you, and proceeds north from there. On this street, more northerly points are further up the screen — until they start to curve in towards the center of the screen, and then start to curve back down, and north is down again.