*hums a quiet tune. twiddles thumbs.*
Are you back? Wasn't that the clearest explanation of the different types of free apps that you've ever read? With that fantastic list of resources at the end? I just wanted to make sure you read it because even I (who have been researching apps for awhile now) found it to be a very helpful breakdown of a very confusing and convoluted topic.
To support the article, here are a few of my personal favorite free apps (I've identified which type of app they are according to the article by including the corresponding number. I won't tease you by posting 4's or 5's since those aren't always free.):
Toca Boca free apps: [2, but only minimal ads to the full versions]
(Toca Tailor Fairy Tales, Toca Kitchen Monsters, Toca Hair Salon Christmas Gift)
These are smaller versions of three of their paid apps, but you can have tons of fun with them without having to buy the paid version and if your kid LOVES the app, then you'll know it's worth the (very small) investment to get the paid version.
This is such a sweet little app about how to handle challenging emotions and it's free. No strings attached.
Geoboard by the Math Learning Center (web app here) This one is free because of a grant. There's a mobile app and also a web app. All free!
Endless Reader This one is free and you can see the first 5 words or so for free, but after that you're required to purchase "word-packs." It's a great way to preview this app and you CAN have fun and enjoy just those first few words on their own, but be forewarned--you and the kids you let play with this app will love it to pieces and you'll quickly realize that the word packs are a bargain and the app is a huge educational goldmine of wonderfulness.
And a few new ones I just discovered for older kids:
Sushi Monster [2, but extremely minimal ads]
This app was created by Scholastic and is promoting their "Fastt Math" curriculum, but the commercial aspect is fairly low-key and the game can be thoroughly enjoyed as a stand-alone game. You need to be able to do addition and multiplication to play these games.
Sound Uncovered and Color Uncovered by the Exploratorium (these are mentioned in the article I linked to, but I wanted to post them here as well) 
Interactive books with fun optical and aural illusions! There's a lot of reading involved in these, so I'd recommend them for older kids or for reading aloud with an adult for younger kids.