You might find some excellent ideas here.
They’re not exactly a series (though if memory serves, some characters do cross from book to book), but Thornton W. Burgess’s books of animal stories were great fun when I was a child. Amazon looks like it has a few nice sets of his work.
And although they are single volumes, both Rudyard Kipling’s Just So Stories and Ernest Thompson Seton’s Wild Animals I Have Known are books that contain animal stories that kids tend to like. The former, with stories such as “How the Camel Got His Hump” and “How the Leopard Got His Spots,” is a little more whimsical and suitable for younger children; the latter is a little more true-to-life (and it must be said, death–while fiction, the stories are fairly realistic depictions of an animal’s life in the wild), and may be more suitable for older children.
And of course, the old standby: Anne of Green Gables, and the subsequent Anne books.
Definitely the Ramona books. I read Henry Huggins when I was a kid, and Beezus and Ramona and Ramona the Pest had come out just before I got too old. My kids both loved the rest of the books. Ramona Forever came out when the older one was in high school, and she still gobbled it up. Every age gets something else out of them, so they are good long termers.
Take a look at some Golden Books, not character based. We had a bunch from when my wife was little, and our kids loved them, and their kids will too when they arrive. I don’t know what is out there now, but our thrift store has a pretty good selection. Some are by Margaret Wise Brown.
Dr. Seuss also, of course.
bump
I got around to the book store this past weekend, and I ended up buying 7 Sandra Boynton books for my SIL and her child. My husband and I looked at so many of the suggestions here (and a few other random ones…did you know the NHL puts out really crappy shapes, letters and numbers books?)
We chose the Boynton ones because we liked the drawings and had a blast reading them to each other! These 7 are all the same size, though the store had other ones in a larger format…those can wait until Christmas!
We got:
[ul]
[li]Blue Hat, Green Hat[/li][li]Doggies[/li][li]Moo, Baa, La La La![/li][li]Horns to Toes[/li][li]Opposites[/li][li]The Going to Bed Book[/li][li]But Not the Hippopotamus[/li][/ul]
My husband was in stitches over the “oops!” in Blue Hat, Green Hat. We still make each other laugh by randomly saying “oops!” to each other!
One of my husband’s cousins invited us to her one-year-old’s birthday, so we also bought *Barnyard Dance! * and the classic Goodnight Moon for her. That kid’s dad is a musician and just crazy enough to actually dance with that book for his daughter, I’m sure! 
Thanks for all the suggestions!
Oohhh, if you like those, did you see Philadelphia Chickens? Children’s music that doesn’t make your ears bleed, with Boynton’s humor and illustrations.
Once the kid gets a little older, I do strongly suggest that you look at the Mr. Putter and Tabby and Poppleton series, both by Cynthia Rylant (but with different illustrators). My almost-three-year-old requests those every single night, but they hold up quite well to rereading compared to most of the other books he likes. (We had to hide some Dora books before we became insane.)
No, I did not see that! I just might try and find it, though! There actually wasn’t that much of a selection, but we went to the book store on the South Shore rather than the ones downtown - the former carried both English and French books while the latter ones tend to mostly only carry English ones, so they have a larger selection.
I figure I got the rest of this kid’s life to buy him or her books (or at least the rest of mine!) so I don’t have to clear off the shelves right away! I’ll see how popular these are first, though! Maybe the kid won’t like chickens (which is a silly thought, but you never know!)
Blue Hat, Green Hat was the first book my youngest could read. 
She memorized it and then learned the words. My nephew used to sit and turn the pages, say “oops!”, and then laugh hysterically. He was about 14 months. Boyton books may be board books but they are loved well beyond the ages that board books are normally put aside.
Good choice.