Look at Sago Mini games. When mine were one, their favourites included Sago Mini Babies, Boats, Space Explorer and Forest Flyer. My three year old still loves them, and my five year old still plays them occasionally too. Also Toca Boca, though those are mostly aimed at slightly older kids. Toca Pet Doctor was a particular favourite when they were younger, though it does require some fine motor control (and is excellent for teaching tap and drag control) . Let me know if you want any more suggestions. It's no coincidence that these are all paid apps. They have no ads or IAP. You can get a lot of them free on Android with Amazon Underground if you have Prime.