My nine year old has an ebook reader because he reads books like the Percy Jackson series (Rick Riordan), Warriors (Erin Hunter), and other long or large book series. I had to put a three-book limit on what could be carried in his backpack because it was so heavy from his books! He enjoys his ebook reader a lot, although he reads paper books too that he gets from the library and from an independent book store in the next building from his guitar lessons.
(Note that his dad takes him to guitar lessons, bought him the Kindle because he was tired of all the space books take up in the house, and continues to buy *and order* books from the same bookstore!
We are both VERY big on supporting small town businesses, but the irony still kills me!

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He (my son) thinks ebook readers are cool. He likes that he can get books almost as soon as he asks for him and that if I pre-order a book, it will show up automatically. He likes that it is easier to carry than two or three books and I think he even likes that he always has his light with him.
He has an iPod too, but I have not installed any reading apps on it. I've been thinking of asking him if he wants one, but he normally just carries both devices around; I like that he can still read his Kindle if his iPod battery runs out.
My husband bought a Kindle for my six year old, but she doesn't know it yet. (She thinks she is borrowing mine.) We have been reading a book on it, and she likes the Kindle, but not that particular book.
She is into The Magic Treehouse right now and has the first 42 of them in paperback. I asked her if she would prefer to finish the series in paper or ebook and she chose paper. As a first grader, she takes those books to school with her and she knows she would not get to take a Kindle; I don't know if that factored into her choice, but I think she may be too young for an ebook reader, which is why I haven't officially turned it over to her. My husband likes to buy gadgets -- but I don't know why he didn't keep this one for himself.
I have a four year old too, but no electronic books for him yet -- I have a Kindle app book on my iPad for him that he likes, but he still prefers sitting with a book in his lap and flipping the paper pages. I'm in the camp that thinks that ebooks readers are best for chapter books with few illustrations, and he is definitely too young for those.