I recently looked around at "baby apps" hoping to find something to launch before letting my one year old play with my phone. (She loves smartphones more than any other object but I don't appreciate the random calendar entries and text messages she sends.) Every single one I found had advertising and half of them had in app purchases. The people making these apps must assume I'm an idiot.
Something like "bubbles" tends to be popular with the younger kids. My kid also was a big fan of "Petting Zoo" (by fox and sheep GMBH) around that time. (It was briefly free in the app store - it's somewhat abstract and interactive with simple gestures.)
The "pat the bunny" app is also pretty accessible for a young kid.
Use "guided access" mode if you want to keep her from switching apps.
As he got older, he enjoyed stuff like the tiny hands sorting games, the Dr Panda apps, "little builders" and other fox and sheep apps, and the lego/duplo stuff. Around three he was also a fan of lara croft go and monument valley, parts of which he could do after watching me. He would build his own lara croft go (and hitman go) scenarios with his playmobil 1-2-3 and magna-tiles.
As far as screen time goes, we settled on weekend (and holiday) access to the device, but we're pretty much making this up as we go.
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.
Thanks for these suggestions (thanks also to dunham in sibling comment). I was having a hard time finding/evaluating the paid options, it's great to have your insights.
> The people making these apps must assume I'm an idiot.
No, they believe (and this is a belief for which there is overwhelming empirical evidence, not an unsupported assumption) that there exist lots of people with low tolerance to up-front costs from whom money can be made via advertising and IAP.
You may not be one of those people, but then you, alone, are insignificant in the market.
