I have him read one book, but I read one to him as well. That gives him some variety but he can concentrate on what he is reading. When he gets older he can read as many as he wants at a time, but until he gets a little better one at a time is enough.
I prefer they read one book at a time, but there is something to be said for mastering the skills required to read more than one book at a time. At school they are required to read certain books for their literacy class and they aren't supposed to read ahead of the assigned pages, etc. so if they only read that one book, they aren't reading enough.
My youngest is a great reader. I am constantly having to remind her that she needs to finsih one "fun" book before she can move onto another. I don't mind if they read a fiction and a non-fiction at the same time, but not 2 of the same kind. I feel like it will hamper their comprehension at this point. My youngest is 5 and has about a 70-75% comprehension...she is doing really well. My oldest knows that she is struggling with her comprehension and hasn't had a hard time following the guidelines I have in place for them. She is 7 and has about 60% comprehension on a good day. She is just one of those kids that is taking a little longer to pick it up. She does great with everything else.
Well I usually have them reading a book that they can read on their own (short) and have them reading some type of history or science story and I'm reading to them that is longer, chapter books. So jumping around is ok as long as they can keep the stories straight. Asking a few comprehension questions before we begin another reading session helps me make sure he's got it under control.
I think I would have a hard time keeping any of my kids to one book. When I think of a child starting to read, I'm thinking simple books that are done in a few minutes. My kids are voracious readers, and I think hampering them by saying they have to read my way is not a good thing. By all means, test for comprehension, we don't want them missing out on that part of the equation. But a child who can comprehend but doesn't read because it's loathesome to him, is no better off than a child who can't read.