We have lots of bookcases. Yes, we have this problem, but I do go through and get rid of things that are broken/ worn/ tattered/ out of date, and things that they have outgrown that are not sentimental.
I purge every so often. I've also stopped buying new ones for the past 8 or 9 months with DD in first and DS still a toddler. When DD was in pre-k & K, we used to buy a lot from the Scholastic book club. I may pick up buying again when DS moves to pre-K. DD and I went to Barnes and Noble during the holidays to specifically find books for her but we came home with like 4 books, 2 of which were travel guides. I didn't want to buy a bunch of random chapter books she might not like. I also don't love taking her to the library because it's book overload and tough to find something age-appropriate, but we'll try again this year.
I like to think we're minimalists. Our house is on the smaller side and I can't stand clutter. DS is 6 (almost 7) and we have mostly moved out of the bulky plastic toy stage. Thank goodness.
But books? The more we own the more DS reads. Sure we go to the library, but he re-reads a lot of what he owns. Sometimes I will find that a series I purchased, say, 6 months ago in hopes that he would read it, is finally found on the shelves and we can't pry his nose out of it. He also goes through books so fast that we really need to have a lot in the house. Yesterday I gave him book 2 of Mac B Kid Spy to read while we were at some medical appointments. 150 pages, some photos, read intermittently between testing, and he was finished with it by 1pm.
We have weeded out most of his baby / toddler books, only keeping a few special selections. He's never been much of a fan of picture books, so we don't have a whole lot of those unless they are reference books. Chapter books? Unless he truly disliked it, we are keeping them ALL for now. We even went as far as getting another bookshelf a few weeks ago.
Goodwill has been amazing for us. That's where I found most of his Diary of a Wimpy Kid, classics, Zombie Goldfish, Big Nate, Harry Potter, and Magic Tree House books. It's also where I've found multiple editions of Calvin and Hobbes for $1-2 each.
As a family we don't spend a lot on clothes or toys, but we spend a LOT on books. And we typically retain what we purchase.