Canabalt
*i.imgur.com/Atwfb2s.jpg
A must have classic. Action packed, minimal gfx, sidescrolling endless runner. The obstacles are procedurally generated, there is a lot of story uncovered the longer you can run.
You can play it right here. Open Source version here.
iTunes
Babylonian Twins
*i.imgur.com/ZE32aLN.jpg
This game has a bit of a backstory, it is made in Iraq in 93 for the Amiga platform. It was never released, and it found a home 15 years later on mobile. You can read more about that here. It is a puzzle platformed, where you have to switch between to twins to cross levels. It is an involving and thought provoking title.
iTunes Play Windows
Polara
*i.imgur.com/QK9Q5eV.jpg
This is a fast paced sidecrolling platformer. You have to switch between red and blue "polarities" or colours to navigate obstacles. You have to choose between same or opposite colour, depending on the situation. So same colour for door, opposite colour for floor. The concept is used in a 3D game on steam called polarity.
iTunes Play Amazon
Perfect Cell
*i.imgur.com/XeozYc2.jpg
This is 2.5 D sidescrolling puzzle platformer. You are a cell escaping a military base. You can choose between assault and stealth strategies. Through the game, you unlock other cells and combine with them, increasing your powers throughout. Graphics are cool and immersive, so you pay attention to the gorgeous backgrounds instead of focusing on the action in the foreground. The controls are made for touchscreens.
iTunes.