While a possibly overrated pick in some aspects, Minecraft is still undoubtedly considered among the greatest open-world games to date. With its release on both iOS and Android a few years ago, player numbers rose dramatically, and a widespread love for the creative, pixelated sandbox was enhanced.
You can build, destroy, forage, fight, and survive in a peculiar world all your own, depending on the mode you choose, including either "creative" or "survival." There is the massive multiplayer cross-platform option as well. You can purchase or download skins and mash-up packs, explore random worlds, and create something out of nothing.
These games offer hours of entertainment and perpetual exploration to lovers of sandbox themes, and better yet, they can be played anywhere and at any time. All of the games mentioned are compatible with iPhone 4s and up, and are adapted to fit the highest possible smartphone graphics capacity, making for a pretty solid gaming experience.