Roguelike and -lite:

• Hoplite — deep movement-basef RL

• micRogue — "chesslike" roguelite

• Pixel Dungeon — a mobile homage to the ideas and mechanics of Brogue

RPG:

• the mobile port of Chrono Trigger — 'nuff said

• Doom & Destiny — Italian RPG, very silly but still deep and satisfying. The free version is the same as the paid, don't bother with the sequel.

Strategy:

• Battle for Polytopia — base game and four civs are free, pay to unlock new civs

• unCiv — FOSS

• Rebuild (and sequels) — everyone's favorite zombie-themed browser game has a mobile port

• Tinyfolks — very satisfying fantasy party management adventure

Management / Plate Spinning:

• Merchant — sell goods to finance adventurers to fight monsters for loot to give to crafters to make goods to sell

• Universal Paperclips — surprisingly deep story from a game about a factory that makes paperclips

• Pikmin Bloom — walk around in real life to visit public art to grow Pikmin sprouts in-game that you send on missions to get materials to grow and buffs more Pikmin sprouts

Some of these cost money, most are free but with limited microtransactions, only Pikmin Bloom is freemium (but the in-game currency is easy to earn through normal gameplay and it's not PvP, so the users who put money in only cheat themselves).