Clash of Clans
Clash of Clans simply has to be on this list, because it's one of the best mobile games period, and has been for over 5 years. I'm one of the millions that have enjoyed this game for over 4 years.
Clash is simple. It's like an offense and defensive tower-defense game with a strategy twist. You build a huge village, upgrade troops and walls, and attack enemies. You can participate in Clan Wars, Clan Games, or start a second village on an island too. I try to avoid most smartphone games, but can't get enough of Clash.
While some would consider this a hardcore game, as it does take a lot of upkeep and constantly battling and grinding for updates, the beauty is that everyone can play it differently. If you want to hop on every once in a while and clear your mines, donate some troops and do a quick battle, you can. Players can be as casual or committed as they want. Just don't get mad when you lose 300-400k from an attack because you didn't check on your village. This is hands-down one of the best games around. So much, in fact, there are well over a dozen knock-offs on the Google Play Store.